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Th e Insti tute Letter

InIstitute foAr AdvancedS Study Summer 2012

The Symplectic Piece How Our Brains Operate: Questions

BY HELMUT HOFER AND DEREK BERMEL Essential to Our Humanness BY JOHN HOPFIELD I can’t understand why people are frightened of new ideas. ll of us who have watched as a I’m frightened of the old ones. Afriend or relative has disap - —JOHN CAGE peared into the fog of Alzheimer’s ar - rive at the same truth. Although we Helmut Hofer, Professor in recognize people by their visual ap - the School of , pearance, what we really are as indi - writes: vidual humans is determined by how ast September, the School our brains operate. The brain is cer - L. of Mathematics launched tainly the least understood organ in its yearlong program with my the human body. If you ask a cardi - Member seminar talk “First ologist how the heart works, she will G

Steps in Symplectic Dynam - give an engineering description of a N I L P I

This image (produced with a Java applet by Alec Jacobson ics.” About two years earlier, pump based on muscle contraction K D R A

at http://alecjacobson.com/programs/three-body-chaos) shows it had become clear that cer - and valves between chambers. If you Y D U

col or ful trackings of the paths of satellites as they evolve from a tain important problems in ask a neurologist how the brain R simple single orbit to a complex multicolored tangle of orbits. dynamical systems could be works, how thinking takes place, From Rudyard Kipling’s “Just So” story solved with ideas coming from well . . . Do you remember Rudyard “The Elephant’s Child” a different , the field of symplectic . The goal was then to bring researchers Kipling’s Just So Stories , full of fan - from the fields of dynamical systems and symplectic geometry together in a program tastical evolutionary explanations, such as the one about how the elephant got its trunk? aimed at the development of a common core and ideally leading to a new field—sym - They are remarkably similar to a medical description of how the brain works. plectic dynamics. The annual meeting of the Society for Neuroscience attracts over thirty thousand regis - Not long before, in my 2010 inaugural public lecture at IAS, “From Celestial trants. It is not for lack of effort that we understand so little of how the brain functions. The Mechanics to a Geometry Based on the Concept of ,” I had described the historical problem is one of the size, complexity, and individuality of the human brain. Size: the human background and some of the interesting mathematical problems belonging to this antici - brain has approximately one hundred billion nerve cells, each con necting to one thousand pated field of symplectic dynamics. The lecture began with a computer program showing others. Complexity: there are one hundred different types of nerve cells, each with its own (Continued on page 12) (Continued on page 8)

Leon Levy Foundation Permanently Endows the Directorship

he Leon Levy Foundation has donated $20 million to the Institute for Advanced of the gift, the Director of the Institute will carry a new title and titular professorship, TStudy. This gift reflects continued support of the Institute by Trustee Shelby White Director and Leon Levy Professor, which will be permanently associated with the posi - and the Leon Levy Foundation, which was created in honor of White’s late husband, tion. This donation brings the capital campaign total to more than $48 million, of the Leon Levy, a leading financier who served on the Institute’s Board of Trustees for fifteen $100 million goal to meet the challenge grant (see page 15). years and as Vice Chairman and President of the Corporation from 1995–2003. The Leon Levy Foundation, founded in 2004 to support scholarship at the highest “This magnificent gift from the Leon Levy Foundation will help to further strengthen level, has continually contributed to the Institute in significant ways. The New Initia - the Institute’s endowment, which is essential to our continued success as a research insti - tives Fund, established by Levy and White in 1998, helped to promote progress in new tution of the highest standard,” said Robbert Dijkgraaf, Director of the Institute and Leon and important programs such as systems biology and theoretical computer , and Levy Professor. “Shelby White’s and the Foundation’s stalwart support has sustained to support emerging research in mathematics and astrophysics. From 2005–12, the important of research at the Institute, including the formation of an archives center Foundation sponsored a Leon Levy Member in the School of Social Science. Past Leon and program, and this most recent donation moves us closer to our $200 million campaign Levy Members have included economists and political scientists who, through the sup - goal. I am honored to be the first to hold the new endowed Directorship.” port of the Foundation, have also delivered an annual Leon Levy Lecture on the nature White, founding Trustee of the Leon Levy Foundation, commented, “My husband of their work while in residence at the Institute. The Foundation funded the new land - would have been proud to have the Institute Directorship named in his honor. He was scaping of the courtyard entrance to Fuld Hall, completed in 2009, as well as other a great believer in the Institute’s mission and work, and he personally devoted many landscape improvements, and it supported a historic landscape study to fully understand hours to its success. The Foundation is pleased that it can continue to support the Insti - the development of the Institute’s campus. A $3.5 million gift from the Foundation in tute’s important initiatives.” 2009 funded the creation of the Shelby White and Leon Levy Archives Center, which This major gift will be increased by $5 million from the $100 million challenge grant is enabling the conservation and collection of the Institute’s current and future hold - made by the Simons Foundation and the Charles and Lisa Simonyi Fund for Arts and ings of records and historical documents, Faculty papers, oral histories, photographs, in 2011 to create a $25 million Leon Levy Endowment Fund. In recognition and other significant documentation. I

Einstein Drive, Princeton, New Jersey 08540 www.ias.edu G12-12800_IAS_SUMMER12.qxp 8/16/12 2:02 PM Page 2

News of the Institute Community

UAN MALDACENA, Professor in the School of Natu - Jral Sciences, has received the 2012 Pomeranchuk Prize our Professors in the School of Natural Sciences––NIMA ARKANI-HAMED, JUAN MALDACENA, of the Institute of Theoretical and Experimental in FNATHAN SEIBERG, and ––have each been awarded $3 million for their significant and Moscow, awarded for outstanding achievement in theoreti - path-breaking contributions to fundamental physics from the Milner Foundation. The inaugural Fundamental cal physics. Maldacena and former School of Natural Physics Prize, created by Internet investor Yuri Milner to recognize transformative advances in the field and inspire Sciences Members LUIS FERNANDO ALDAY (2007, interest in fundamental physics, was awarded to five additional recipients––former Members 2008–10) and AMIT SEVER (2011–12), along with Pedro (1992–93 and 2002), Professor at the Institut des Hautes Études Scientifiques, and Ashoke Sen (1996–98), Professor Vieira, were awarded the Journal of Physics A Best Paper Prize at the Harish-Chandra Research Institute; Alan Guth of the Massachusetts Institute of Technology; Alexei Kitaev 2012 for their paper “Y-system for Scattering Amplitudes” of the California Institute of Technology; and Andrei Linde of Stanford University. The physicists will serve on the in Journal of Physics A: Mathematical and Theoretical 43 selection committee for future recipients of the prize, which will be awarded annually. “These exceptional prizes are (2010). Alday is Professor at the Mathematical Institute of well-deserved recognition of the power of visionary ideas by daring individuals,” said Robbert Dijkgraaf, Director the University of Oxford, and Sever is a Senior Postdoctoral of the Institute and Leon Levy Professor. “The Institute has a remarkable legacy in advancing our understanding of Fellow at the Perimeter Institute for Theoretical Physics. the fundamental laws of nature, and the work of these four current Faculty members illustrates the current strength q and excellent future prospects of research in theoretical physics at the IAS. We wish to compliment Yuri Milner for OAN WALLACH SCOTT , Harold F. Linder Professor his extraordinary generosity that we hope stimulates new generations of physicists.” Jin the School of Social Science, and KAREN UH - LENBECK , Robert and Luisa Fernholz Visiting Professor (2012) in the School of Mathematics, have been awarded of large drawings on the medieval epic The Song of the Ni - nscriptiones Atticae Euclidis anno posteriores. Ed. tertia. Pars honorary degrees by . belungen , examining the epic’s course through German his - I I: Fasc. 5: Leges et decreta annorum 229/8–168/7 by q tory and the artist’s biography as well as the place the STEPHEN TRACY, long-term Visitor in the School of His - drawings occupy in the art, culture, and politics of Germany. torical Studies, and Voula N. Bardani has been published by ICHARD TAYLOR, Professor in the School of Math - Walter De Gruyter. The work, which publishes 325 laws and ematics, and BRUCE KOVNER, an Institute Trustee q R decrees from the first year of independence from Macedon - and Chairman of Caxton Alternative Management LP, ale University Press has published In God’s Shadow: ian occupation and the end of the third Macedonian War, have been elected to the American Academy of Arts and Politics in the Hebrew Bible by MICHAEL WALZER, Y was done under the auspices of the Inscriptiones Graecae Sciences, along with eight former Institute Members and Professor Emeritus in the School of Social Science. project of the -Brandenburg Academy of Sciences. Visitors. Drawing on decades of thinking about Biblical politics, q Walzer discusses the views of the ancient biblical writers q UBHANKAR BANERJEE, a former Director’s Visi - AROLINE WALKER BYNUM , Professor Emerita in on justice, hierarchy, war, the authority of kings and tor (2011), is the recipient of a 2012 Lannan Cultural the School of Historical Studies, has been elected to priests, and the experience of exile. S C Freedom Award from the Lannan Foundation. Banerjee is the Orden Pour le Mérite für Wissenschaften und Künste q a photographer and writer and the editor of Arctic Voices: of the Federal Republic of Germany. Bynum is one of two VISHAI MARGALIT , former George F. Kennan Resistance at the Tipping (Seven Stories Press, 2012). new members elected by the membership to the Order, A Profe ssor (2006–11) , in the School of Historical q which has a total of thirty-seven German and thirty-six Studies, has been awarded the Ernst Bloch Prize by the foreign members. city of Ludwigshafen by the Rhine, which recognizes AXIM KONTSEVICH, former Member (1992–93) q works important to a critical examination of the pres - Min the Schools of Natural Sciences and Mathemat - ics and Visitor (2002) in the School of Mathematics, has erghahn Books has published Myth and Modernity: Bar - ent. Margalit is now an Honorary Fellow at the Van Leer been awarded the 2012 Shaw Prize in Mathematical Sci - lach’s Drawings on the Nibelungen by PETER PARET , Institute. B ences for his work in algebra, geometry, and mathemati - Professor Emeritus in the School of Historical Studies, and q cal physics. Kontsevich is Professor at the Institut des Helga Thieme. The book addresses Ernst Barlach’s sequence OBBERT DIJKGRAAF, Director of the Institute and Hautes Études Scientifiques. R Leon Levy Professor, has been made a Knight of the q Order of the Netherlands Lion. Dijkgraaf, who is Distin - OANNE MANCINI, former Visitor (2009–10) in guished University Professor of at Contents the School of Social Science, has received the 2011 the University of Amsterdam and past President of the J Patricia and Phillip Frost Essay Award from the editorial Royal Netherlands Academy of Arts and Sciences 2 News of the Institute Community board of American Art for her article “Pedro Cambón’s (2008–12), also has been appointed as an honorary 3 Major Starr Foundation Grant Supports Simons Asian Objects: A Transpacific Approach to 18th-Century member of the Netherlands Physical Society and the Center for Systems Biology California” (Spring 2011). Mancini is Lecturer at the Royal Netherlands Chemical Society. Peter Goddard: Thoughts on Stepping Down as Director National University of Ireland, Maynooth. 4 Pre- Higgs Celebration at IAS q q 5 Leibniz, Kant, and the Possibility of Metaphysics RAHAM FARMELO, a Director’s Visitor, has been (and Some Ado about Nothing) he 2012 Raymond and Beverly Sackler Prize in the awarded the 2012 Medal and Prize from the 6 Modular : Driven by Inherent Beauty and G TPhysical Sciences has been awarded to SARA Human Curiosity Institute of Physics for his outstanding work in commu - SEAGER , former Member (1999–2002) in the School of nicating science to a broad audience, in particular for his 9 Gathering Arctic Voices at the Institute, with the Woods Nat ural Sciences, for her analysis of the atmospheres and to Inspire book The Strangest Man: The Hidden Life of Paul Dirac, internal compositions of extrasolar planets. Seager is 10 Liberal Democratic Legacies in Modern Egypt: The Role Mystic of the Atom (Basic Books, 2009), which he worked Class of 1941 Professor of Physics and Planetary Science of the Intellectuals, 190 0–1950 on during his stay at the Institute. at the Massachusetts Institute of Technology. 13 Finding Structure in Big Data q q 14 Andrew W. Mellon Foundation Provides Critical USSELL IMPAGLIAZZO, Visiting Professor Support for Assistant Professors he Norwegian Academy of Science and Letters has R(2007 –12) and former Member (2003) in the School awarded the 2012 Abel Prize to ENDRE SZEMERÉDI, Daniel H. Saracino Endowment Supports Visiting T Professorship in School of Mathematics of Mathematics, is among twenty-one , former Member (2007–08, 2009–10) in the School of theoretical physicists, and theoretical computer scientists 15 Benedict H. Gross Elected to the Board of Trustees Mathematics, for his contributions to discrete mathematics selected as Simons Investigators in 2012. The program, The Campaign for the Institute: Friends Bolster and theoretical computer science, and their impact on Campaign to Raise Capital Funds which is supported by the Simons Foundation and is in its additive theory and ergodic theory. Szemerédi is inaugural year, is intended to enable outstanding scientists State of New Jersey Professor of Computer Science at Questions and comments regarding the Institute Letter should be to undertake long-term study of fundamental questions. Rutgers, the State University of New Jersey, and Research directed to Kelly Devine Thomas, Senior Publications Officer, via Impagliazzo is Professor at the University of California, San Fellow at the Alfréd Rényi Institute of Mathematics, email at [email protected] or by telephone at (609) 734-8091. Diego. Eight former Institute Members and Visitors were Hungarian Academy of Sciences, Budapest. Issues of the Institute Letter and other Institute publications are also selected for the first class of Simons Investigators. q available online at www.ias.edu/about/publications. q TEVEN VANDERPUTTEN, former Member (2005) Articles from the Institute Letter are available online at ÁSZLÓ LOVÁSZ, Visiting Professor (2011–12) in Sin the School of Historical Studies, has received the www.ias.edu/about/publications/ias-letter/articles. L the School of Mathematics, and three former Friedrich Wilhelm Bessel Research Award of the Alexan - To receive monthly updates on Institute events, videos, Institute Members have been elected to the National der von Humboldt Foundation in recognition of his work and other news by email, subscribe to IAS eNews at Academy of Sciences. Lovász is Professor at Eötvös on the social and cultural history of the Western Middle www.ias.edu/news/enews-subscription. Loránd University in Budapest. Ages. Vanderputten is a Professor at Ghent University.

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Major Starr Foundation Grant Supports Simons Center for Systems Biology

$7.5 million grant from the Starr Foundation to the That is what our grantmaking is about: fostering the damental biological problems. Leibler and his colleagues ASchool of Natural Sciences’ Simons Center for Sys - exploration and development of new opportunities.” are developing a theoretical understanding of biological tems Biology will contribute to the creation of a perma - In the last decade, the Simons Center has enabled a functions—in one set of studies, they are investigating nent endowment to cover the core costs of the center’s number of outstanding scientists to move into the fore - patterns of gene evolution that suggest how different program and secure its position as a leading center for re - front of the field and make an indelible impact at lead - parts of proteins interact. search and postdoctoral training in quantitative biology. ing academic institutions and research laboratories Also at the Simons Center, Faculty and Members are The grant completes the of two challenge around the world. The center was established in 2005 by actively engaged in a number of areas of cancer research, grants from the Simons Foundation to establish a per - Arnold J. Levine, now Professor Emeritus in the School including identifying the role genes play in the origins of manent endowment; additionally, it counts toward the cancers and the metabolism of cancer cells, developing $100 million capital campaign challenge made by the strategies to profile specific cancers, and exploring Simons Foundation and the Charles and Lisa Simonyi At the Simons Center, Faculty and means to improve cancer prevention and treatment Fund for Arts and Sciences in 2011. The Institute will Members are actively engaged in a strategies. Studies also include path-breaking analyses of add $7.5 million from the challenge grant to create a $15 number of areas of cancer research, viruses, in particular the influenza genome and evolu - million endowment to support a named Professorship tionary changes that are the result of selection pressures and three Memberships in biology. including identifying the role genes play in both virus and host. Additionally, the center is “We are incredibly grateful to the Starr Foundation in the origins of cancers and the engaged in a four-year collaborative study, “Autism and for its generous endorsement of the Institute’s work in metabolism of cancer cells, developing Single Nucleotide Polymorphisms in the IGF Pathway,” theoretical biology,” said Robbert Dijkgraaf, Director of which investigates the frequency of Single Nucleotide the Institute and Leon Levy Professor. “This grant will strategies to profile specific cancers, and Polymorphisms (SNPs) in selected genes that populate enable the Institute to continue to attract the best sci - exploring means to improve cancer the interrelated signal transduction pathways in individ - entists in the field, advance research into today’s biolog - prevention and treatment strategies. uals with autism spectrum disorder. ical complexities, and chart new territory.” Currently, two distinguished biologists, John Hopfield The grant complements the Starr Foundation’s exist - of Princeton University and Albert Libchaber of the ing far-reaching achievements in supporting biomedical of Natural Sciences, and was named for James H. Rockefeller University, are Visiting Professors at the research. “Research in the fields of genomics, stem cells, Simons, a Trustee of the Institute and former Member Simons Center and are actively working with Members and neuroscience has increased in complexity, but that (1972) in the School of Mathematics, and his wife, Mar - and others on key biological problems related to the complexity creates opportunities,” said Maurice R. ilyn Hawrys Simons. , a Professor in the dynamics of complex systems (see article, page 1). The Greenberg, Chairman of the Starr Foundation. “It is School of Natural Sciences who holds a faculty position establishment of a permanent endowment for the exciting to see the collaborations among biologists, jointly at the Institute and the Rockefeller University, is Simons Center will allow it to support the research of physicists, mathematicians, and chemists as they forge working to extend the interface between physics and three permanent Professors and a total of up to fourteen new paths to understanding diseases and possible cures. biology to create new solutions and approaches to fun - Members or Visiting Professors. I

Peter Goddard: Thoughts on Stepping Down as Director The following text is excerpted from remarks given by Peter Helen and I ourselves received when I first came to the Insti - Goddard, the Institute’s eighth Director (January 2004–June tute as a Member in 1974, and, even more, when I became 2012), at a dinner with Trustees, Faculty, and others in May. Director. Trustees, Faculty, Staff, and Members all made us Peter Goddard first came to the Institute as a Member in the feel really at home. And that very special , the Friends School of Natural Sciences when he was twenty-nine in 1974. of the Institute, welcomed us into the wider community. He returned to the Institute in 1988 as a Member in the School Rachel [then Associate Director of the Institute] of Mathematics. A mathematical physicist, he is distinguished took us to meet Mary Keating, a founding member of the for his pioneering contributions in the areas of string theory, Friends, who entertained us to tea. And, she very kindly quantum field theory, and conformal field theory. Upon step - rang Rachel the next day to say how much she had ping down as Director at the end of June, he became a Profes - enjoyed meeting us, but there was one point about which sor in the School of Natural Sciences. she was uncertain: she was not sure which one of us it was that would be becoming Director of the Institute. I am n some ways it seems like yesterday that the telephone sure you all realize just how much I have depended on Irang in Cambridge and Steve Adler asked me whether I Helen’s help, and what she has done for the Institute. would be interested in being considered in the search for I The last eight years have seen some challenging the next Director of the Institute. When I told my wife times because of the global economic crisis, which has Helen, she said that it was an enormous honor to be ap - affected us all in so many ways. But, I believe, the essentials proached in that way. And then a little while later, she said, of the academic life of the Institute have not been dam - “But, you are not taking it seriously, are you?” aged. First, this is because of the extremely generous sup - I When Lewis Strauss, who was chairing the search port provided by the Institute’s Trustees. The Institute’s committee in 1946, consulted about the founding Director, Abraham Flexner, told the Trustees, at Y

qualities he should be seeking in the third Director of the their first meeting in 1930, “it is not the function of a board V E L . F -

Institute, Einstein told him he should look for a quiet man of trustees to be merely amiable.” Nobody could accuse our J E G R

who would not disturb people who are trying to think. I Trustees of being “merely amiable”! Their commitment to E S have tried to satisfy the Einstein criterion. the Institute continues to be quite extraordinary: a really Peter Goddard served as the eighth Director of the Institute, I Over the last eight and a half years, as I have welcomed great institute deserves really great trustees and vice-versa. from January 1, 2004, through June 30, 2012. each incoming group of new Members on behalf of the I Also absolutely essential to our safe passage through Institute, I have remarked on the warmth of the welcome these extraordinary economic times has been the way the here and take on the Institute. Faculty and the Staff have worked together to reduce I And the other gift, which I also value enormously, is expenditure and increase income, all this without any to be able to stay here and work in the School of Natural major adverse impact on the Institute’s work and life. For Sciences, as a part of this community, without administra - this, I am enormously grateful to them—the last thing I tive duties. I am looking forward to irresponsibility! would have wished to preside over would have been a I But, that said, I shall really miss working with my decline because of financial stringency in the freedoms colleagues in the Director’s office and throughout the and opportunities that the Institute provides. Institute. The administration here is focused, like no other I The Trustees have already given me the best possi - institution I have known, on supporting the work of the ble presents. Nothing could be a better gift as I step down Faculty and Members and not the theory and practice of www.facebook.com/InstituteforAdvancedStudy www.twitter.com/the_IAS as Director than to be able to hand on the Institute to administration for its own sake. Every year, the reports of

E www.youtube.com/videosfromIAS someone as outstanding as Robbert Dijkgraaf. My con - the Members make this very evident. And I have felt T I H

W gratulations and thanks go to the search committee for deeply privileged to be Director of an institution with E C U

R persuading Robbert, his wife Pia, and their family to move these ideals and ethos. I B 3 G12-12800_IAS_SUMMER12.qxp 8/16/12 6:06 PM Page 4

Pre-Dawn Higgs Celebration at IAS

of the factors that made its detection so difficult. BY GRAHAM FARMELO By the early 1990s, a combination of robust theoretical Graham Farmelo is a By-Fellow at Churchill College, arguments and the well-attested experimental results, Cambridge. He wrote much of his biography of Paul n Wednesday, July 4, shortly after 4 a.m., the Insti - notably from Stanford and CERN, persuaded many parti - Dirac, The Strangest Man: The Hidden Life of Paul Otute’s new Director, Robbert Dijkgraaf, was in cle physicists that the Higgs boson exists. Yet, by 2007, Dirac, Mystic of the Atom (Basic Books, 2009), and Bloomberg Hall, cracking open three bottles of vintage some theoreticians started to get cold feet and a few even of his next book, on Winston Churchill and his nuclear champagne to begin a rather unusual party. He was among began to wonder if the Standard Model might be about to physicists, as a Director's Visitor at the Institute. the scientists who had been in the Hall’s lecture theater crumble. Arkani-Hamed was having none of this, which since 3 a.m. to watch a presentation from Geneva on the is why he made the bet. A member of the public took him that he would win his bet: “All decent theorists have latest results from the CERN laboratory’s Large Hadron up on it, after a fashion, wagering $100 plus an amount known for some time that there had to be a Higgs boson Collider. In the closing moments, after CERN’s Director- proportional to the mass of the Higgs, if it showed up. The more or less where it was found.” After a glass of cham - General Rolf Heuer cautiously claimed the discovery of a amount was one dollar for each unit of the Higgs’s mass, as pagne, he added: “Nature is extremely constrained— new sub-atomic particle —“I think we have it, yes?” —ap - measured in the physicists’ preferred unit of GeV/c 2, a tad there aren’t nymphs around every tree and dryads around plause broke out in the CERN auditorium and in the less than the mass of the lightest hydrogen atom. every corner. Physics works.” Bloomberg Hall lecture theater. Within minutes, the IAS Experimenters at the Large Hadron Collider The party was probably unique in the history of the party was underway. announced last December that they had found the first Institute—no one could remember another occasion The new particle shows several signs that it is the signs of a particle consistent with the Higgs boson, with when its physicists quaffed champagne and speared Ital - Higgs boson, the only missing piece of the Standard Model, about 125 GeV/c 2. It remained to be seen whether the ian-style macerated strawberries after a seminar begin - which gives an excellent account of nature’s electromag - particle really did exist, or if the first hint had been a ning in the middle of the night. Renée Hlozek, an netic, weak, and strong interactions. Although some glitch in the data. A few months ago, when the CERN astrophysicist at Princeton University, brought cookies physicists had come to doubt whether the boson existed, authorities announced that two of their experimental decorated with a symbol denoting the Higgs field. On Nima Arkani-Hamed, Professor in the Institute’s School groups would announce their results jointly on July 4, loan from Marilena LoVerde, a Member in the Institute’s of Natural Sciences, was so confident that in 2007 he bet the rumor mill began to grind noisily, especially on the School of Natural Sciences, and Laura Newburgh, a a year’s salary that it would be detected at the Large Internet. A week earlier, it was common knowledge physicist at Princeton University, was a diorama of Hadron Collider. In the week before the CERN presenta - among physicists that something dramatic was afoot. CERN’s huge ATLAS particle detector, delicately craft - tion, Arkani-Hamed invited colleagues to the party and Arkani-Hamed expected between ten and fifteen col - ed from Peeps (marshmallow candies with hardened organized the catering. Convinced that he had won his leagues to attend the screening of the presentation, but sugar shells), as well as paper and items molded from bet, he bought three bottles of champagne, including two his prediction was not one of his most accurate—he was wax, held together with sticky tape. of Special Cuvée Bollinger. too low by a factor of three. The physicists watched A few hours after the party broke up, Robbert Dijk - The Higgs boson has played a central role in fundamen - mostly in silence as the experimenters’ story unfolded— graaf tweeted a photograph of the remnants of the cele - tal physics since 1964, when several theoreticians—includ - the particle appeared to exist and to decay into others, bration: “Decay products of 3 am #higgs event . .. Never ing Peter Higgs at the University of Edinburgh—suggested such as pairs of , at the expected rates (though forget 1st week as Director!” By then, Arkani-Hamed an explanation of why most fundamental particles have with one tantalizing exception). Unusually for a seminar had talked with the New York Times ’s science correspon - mass. According to the theory, these particles acquire their like this, the IAS audience —as well as the one in dent Dennis Overbye and given him the quote that con - mass from their interactions with the Higgs field, invisible CERN—broke into applause several times and there cluded his report on the Higgs-like particle: “It’s a and elusive but permeating the entire universe. As with all were even a few gasps of delight. The same was happen - triumphant day for fundamental physics. Now some fun quantum fields, there is a particle associated with the Higgs ing all over the world, now that the quest for the Higgs begins.” field—the Higgs boson, which has no electrical charge, no boson was apparently coming to fruition. Soon afterward, Arkani-Hamed heard that the gen - spin, and a mass that is unfortunately not possible to calcu - Arkani-Hamed later said that he “never doubted for a tleman who bet against Higgs agreed to pay him his win - late. This uncertainty about the mass of the particle is one second” that the Higgs particle would be detected and nings, $225. The check is, apparently, still in the mail. I

In the pre-dawn of July 4, his fourth day as the ninth Director of the Institute, Robbert Dijkgraaf (top center and bottom left with School of Natural Sciences Professors Nima Arkani-Hamed, seated, and Edward Witten, standing, Chiara Nappi, left, and Member Marilena O L E

M LoVerde, right) opened champagne in Bloomberg Hall for R A F a celebration of CERN’s discovery of a Higgs-like boson. M A H

A Members witnessed the announcement in the lecture hall R G

S and later celebrated with macerated strawberries, Higgs- O T

O denoted cookies (bottom right), and a Peeps-populated dio - H P L

L rama of CERN’s ATLAS particle accelerator (top right). A

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Leibniz, Kant, and the Possibility of Metaphysics (and Some Ado about Nothing)

that our freedom could never be cognized. Yet, all is not lost, for Kant believed that the BY BRANDON C. L OOK existence of God, the immortality of the soul, and the freedom of the will must be con - sidered as postulates of morality and not objects of knowledge. In this sense, for Kant, f the eighteenth century is to be seen as the “Age of ,” then one of the crucial sto - there is a primacy of practical philosophy (ethics) over theoretical philosophy. Iries to be told is of the trajectory of philosophy from one of the most ardent proponents While all previous philosophers were, in Kant’s mind, guilty of various errors, Leibniz of the powers of human reason, Gottfried Wilhelm Leibniz (1646–1716), to the philosopher occupied a special position in his conception of the history of philosophy and the histo - who subjected the claims of reason to their most serious critique, (1724– ry of reason’s pretensions. Indeed, according to Kant, Leibniz’s metaphysical system was 1804). Not only is the story of Kant’s Auseinandersetzung with Leibniz important historically, “an intrinsically correct platonic concept of the world.” What exactly was the problem? it is also important philosophically, for it has implications about the nature and possibility In a short chapter of the Critique of Pure Reason , “On the Amphiboly of Concepts of of metaphysics, that branch of philosophy concerned with fundamental questions such as Reflection,” Kant makes explicit this disagreement. Put briefly, what there is, why there is anything at all, how existing things are causally connected, and Kant believed that Leibniz failed to undertake a real cri - how the mind latches onto the world. Like many philo - tique of the powers of the human mind, failed to see the sophical debates, however, it is also prone to a kind of fundamental distinction between sensibility and “eternal recurrence” to those who are ignorant of it. understanding, considered perceptual experience a Leibniz was a “rationalist” philosopher; that is, he kind of conceptual confusion, and ultimately mis - was committed to two theses: (i) he believed that the took appearances for things in themselves. More mind has certain innate ideas—it is not , as exactly, Kant argued that Leibniz’s metaphysical and his fellow empiricists say, a tabula rasa or blank slate; system is the consequence of the fallacy of and (ii) he believed in—and, in fact, made explicit— “amphiboly”—a fallacy that arises from ambigui - the “principle of sufficient reason,” according to which ties in grammatical form. Consider the following “there is nothing for which there is not a reason why it Marxist claim: “Last night I caught a prowler in is so and not otherwise.” This principle had enormous my pajamas. What he was doing in my pajamas, I metaphysical consequences for Leibniz, for it allowed don’t know.” The joke, of course, turns on the him to argue that the world, as a of contingent ambiguity in the phrase “in my pajamas”—does it things, could not have the reason for its existence with - modify the speaker or the prowler? In a somewhat sim - Immanuel Kant in it; rather there must be an extramundane reason— ilar way, Kant argued that because Leibniz failed to dis - God. Further, as a response to the mind-body problem, tinguish the faculties of sensibility and understanding and Leibniz advanced the theory of “pre-established harmony,” according to which there is no to recognize the unique contribution of each in judg - Gottfried Wilhelm Leibniz interaction at all between substances; the mind proceeds and ment, he attributed concepts “unfolds” according to its own laws, and the body moves accord - While all previous philosophers appropriate to objects of the understanding to objects of sensi - ing to its own laws, but they do so in perfect harmony, as is fit - were, in Kant’s mind, guilty of bility and thereby made fallacious arguments. For example, Kant ting for something designed and created by God. Strictly saw an amphibolous use of the concepts of comparison of iden - speaking, however, Leibniz was not a dualist; he did not believe various errors, Leibniz occupied a tity and difference, which he argued allowed Leibniz to falsely that there were minds and bodies—at least not in the same sense special position in his conception of extend beyond its legitimate scope the principle of the identity and at the most fundamental level of reality. Rather, in his of indiscernibles (if two things are qualitatively identical, that is, mature metaphysical view, there are only simple substances, or the history of philosophy and the indiscernible, then they are numerically identical). According monads , mind-like beings endowed with forces that ground all history of reason’s pretensions. to Kant, Leibniz, comparing the concepts of two individuals with phenomena. Finally, according to Leibniz, since these simple sub - each other, argued that they must be distinct, that is, discernible; stances are ontologically primary and ground the phenomena of matter and motion, and he then extended this principle to objects of the senses and claimed to have made a and time are merely the ordered relations derivative of the corporeal phenomena. Leibniz con - great discovery about nature: “there are never in nature two beings that are perfectly trasted his view with that of Isaac , according to whom there is a sense in which space alike.” But objects of the senses, objects found in nature, must first be given to us in space. and time can be considered absolute and space can be considered something substantial. Therefore, “the difference in place already makes the multiplicity and distinction of In his Critique of Pure Reason (1781; 2nd ed. 1787), Immanuel Kant presented a revo - objects as appearances not only possible in itself but also necessary.” lutionary philosophical view, one that challenged rationalist and empiricist orthodoxy and The issues at the center of the debate between Leibniz and Kant are still with us— one that, he believed, provided answers to certain questions that had been the subject of though they sometimes manifest themselves in different ways. For example, in a recent perennial conflict. Kant advocated “transcendental idealism,” according to which space book, A Universe from Nothing: Why There Is Something Rather than Nothing (Free Press, and time are forms of sensibility—that is, everything that is given to our minds is given as 2012), the physicist Lawrence Krauss takes up what Leibniz considered the fundamental an object in space and time—and there are certain (innate) pure concepts of the under - question of metaphysics. Krauss proposes that normal “material” objects of the universe standing that make experience possible. For example, Kant agreed with the empiricist are manifestations or arrangements of quantum fields and that it is possible for a quan - , who had argued that the idea of cause or necessary connection cannot be tum field to be arranged in such a way that no ordinary physical objects exist, but that traced to an immediate sense impression; yet he believed that the concept of cause must this state is deeply unstable and most likely to go from a “null-object” state to a state with be in our minds in order for us to experience the world at all. A proposition expressing objects. Hence, something from nothing. Writing recently in the New York Times , the universality and necessity (as in claims of causation) is objectively valid not because of philosopher David Albert raises some pointed questions for Krauss: Where do the quan - what we experience out there but because of the way our minds function. Kant also argued tum fields come from? Why are the laws of nature what they are? And, still, why is there that there is an important distinction to be drawn between the appearances, or phenome - a world of quantum fields at all? None of these questions are answered by Krauss, and na, and things in themselves, or noumena . According to Kant, we can only have knowl - without answers to these questions it would seem philosophically presumptuous to claim edge ( Erkenntnis ) of things in the phenomenal realm or of the conditions for the to have answered the fundamental question of metaphysics. But even Albert’s questions possibility of experience; things outside of space and time or beyond the bounds of sense— themselves presuppose Leibniz’s principle of sufficient reason—that there is a reason for the “supersensible”—are off limits for our knowledge claims. Thus, in Kant’s view, meta - everything. For his part, however, Kant had argued that the principle of sufficient reason physics as a science of the supersensible as Leibniz and others had conceived it is doomed is valid only in the realm of experience and that it does not even have meaning when to failure; but a metaphysics that first establishes the boundaries of our knowledge claims applied to things outside the world of sense (and prior to the origin of the world). In and determines the conditions for the possibility of experience may indeed succeed. other words, the proper metaphysical view is that natural science, conceived as an induc - The importance of the Critique of Pure Reason cannot be disputed. This work funda - tive and deductive enterprise based on sense experience, can never give us an answer to mentally changed the way philosophers think of the big-ticket items of metaphysics— the question why is there something rather nothing? But neither can rationalist metaphysics. the existence of God, the immortality of the soul, the freedom of the will. Leibniz In the end, Kant counsels philosophical humility. 1 I believed that we can know that God exists, that the soul is immortal, and that there is freedom (though in a mitigated sense, and here Leibniz is aware that certain aspects of 1 But who wants to be humble? Consider the following concluding argument. Despite its tendency to pretentious nonsense, his system lead him into trouble on this topic). Kant, on the other hand, believed that (1) Metaphysics is better than nothing. we have theoretical knowledge of none of these things, and in the Critique of Pure Rea - Colleagues in the School of Natural Sciences, however, certainly believe that son famously showed how all arguments for the existence of God are flawed, that the (2) Nothing is better than empirical science. received views about the simplicity and immortality of the soul were problematic, and Colleagues in the School of Mathematics, meanwhile, surely believe in the transitivity relation: if a > b and b > c, then a > c. (3) Therefore, metaphysics is better than empirical science. Brandon C. Look is University Research Professor in the Department of Philosophy at Despite the apparently rock-solid deductive form of this argument, colleagues in the School of Social the University of Kentucky. He spent the 2011–12 academic year at the Institute as the Science likely see in it merely disciplinary posturing. As a philosopher, however, I must sadly (and humbly) admit that, even if the conclusion is true, the argument itself commits the fallacy of amphi - Hans Kohn Member in the School of Historical Studies, where he worked on a book boly, for it trades on the ambiguous meaning and grammar of “nothing” and expressions involving it. I manuscript “Leibniz, Kant and the Possibility of Metaphysics.” suspect that any proposed answer to the question “Why is there something rather than nothing?” will flirt with other ambiguities attached to nothingness.

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Modular Arithmetic: Driven by Inherent Beauty and Human Curiosity

it is more than a curiosity. I hope this article will con - where a, b, and c are themselves whole such BY RICHARD TAYLOR vince her that it is. that a, b, and c have the same parity (i.e., are both odd An important observation is that any arithmetic or both even). odular arithmetic has been a major concern of equality that is true in normal arithmetic is also true in But what if we change the problem slightly? What Mmathematicians for at least 250 years, and is still a any whole number you like. about asking for the solution of very active topic of current research. In this article, I will This easily results from the observation that one can explain what modular arithmetic is, illustrate why it is of wind the usual number around the modular clock X 2 + Y 2 = 2 Z 2 importance for mathematicians, and discuss some recent face, turning usual arithmetic into modular arithmetic. or breakthroughs. ______X 2 + Y 2 = 3 Z 2 For almost all its history, the study of modular arith - metic has been driven purely by its inherent beauty and o illustrate a major reason why mathematicians care in whole numbers? It turns out that as soon as we find by human curiosity. But in one of those strange pieces of Tabout modular arithmetic, let me start with one of one non-zero solution in whole numbers, the method serendipity which often characterize the advance of the oldest questions in mathematics: Find Pythagorean described in Euclid’s Elements applies, and we can human knowledge, in the last half century modular triples, i.e., find whole number solutions, to the equation describe all solutions in whole numbers explicitly. So for arithmetic has found important applications in the “real instance the equation world.” Today, the theory of modular arithmetic (e.g., X 2 + Y 2 = Z 2. Reed-Solomon error correcting codes) is the for X 2 + Y 2 = 2 Z 2 the way DVDs store or satellites transmit large amounts By Pythagoras’s theorem, this is the same as finding of data without corrupting it. Moreover, the crypto - right-angled triangles all of whose sides have lengths has a solution graphic codes which keep, for example, our banking that are whole numbers (when measured in the same transactions secure are also closely connected with the units). For instance X = 1 Y = 1 Z = 1 theory of modular arithmetic. You can visualize the usual arithmetic as operating on 32 + 4 2 = 5 2 and one deduces that the general point with whole num - points strung out along the “number line.” ber coordinates is of the form and there is a right-angled triangle: X = ( a2 + 2 ab – b2 )c/2 Y = (– a2 + 2 ab + b2)c/2 Z = ( a2 + b2)c/2

To add 3 and 5, you start at 0, count 3 to the right, and where a, b, and c are themselves whole numbers such then a further 5 to the right, ending on 8. To multiply 3 that a, b, and c have the same parity. by 5, you start at 0 and count 3 to the right 5 times end - However, if you search for non-zero whole number ing up at 15. These sorts of operations should be famil - solutions to X 2 + Y 2 = 3 Z 2, you won’t find any. What iar from elementary school. is the difference between X 2 + Y 2 = Z 2 or X 2 + Y 2 In modular arithmetic, one thinks of the whole num - = 2 Z 2, and X 2 + Y 2 = 3 Z 2? The answer comes from bers arranged around a , like the hours on a clock, The 3,800-year-old Babylonian tablet Plimpton 322 modular arithmetic. instead of along an infinite straight line. One needs to lists Pythagorean triples. The second column of the Suppose there was a solution to decide at the outset how many “hours” our clock is going tablet lists values for X and the third column the corre - to have. It can be any number, not necessarily 12. As a sponding value of Z; the value of Y is not listed. X 2 + Y 2 = 3 Z 2, first illustration, let’s suppose that we have seven “hours” on our clock—we say we are doing arithmetic modulo 7. with X, Y, and Z non-zero whole numbers. We can arrange that no whole number bigger than 1 divides all MODULO 7 of X, Y, and Z. (If it did, simply divide each of X, Y, and Z by this common factor, and they still form a solution to the same equation. If need be, we repeat this process. Note that as the numbers X, Y, and Z get smaller in absolute value each time, but remain whole numbers, this procedure must eventually stop.) Then there would be a solution to the same equation in arithmetic modu - lo 3. But in arithmetic modulo 3 we have

In modern notation, the solutions listed on Plimpton 3 × Z 2 ≡ 0 × Z 2 ≡ 0 mod 3 322 are as follows: and

02 ≡ 0 and 12 ≡ 1 and 22 ≡ 1 mod 3,

To add 3 and 5 modulo 7, you start at 0, count 3 clock - i.e., wise, and then a further 5 clockwise, this time ending on 1. To multiply 3 by 5 modulo 7, you start at 0 and count X 2 ≡ 0 or 1 mod 3 and Y 2 ≡ 0 or 1 mod 3. 3 clockwise 5 times, again ending up at 1. We would write Richard Taylor, who became a Professor in the School 3 + 5 ≡ 1 mod 7 and 3 × 5 ≡ 1 mod 7. of Mathematics in January, is a leader in number the - ory who, with his collaborators, has developed power - As we mentioned above, there is nothing special about It is noteworthy that some of these solutions are quite ful new techniques that they have used to solve 7. We can put any number of “hours” around our clock complicated, but we don’t know how they were generat - important long-standing problems. With Andrew face and do arithmetic modulo any whole number. Our ed. Could it have been trial and error, or did the Baby - Wiles, he developed the Taylor-Wiles method, which usual clocks can be used to do arithmetic modulo 12. If lonians know an algorithm? they used to help complete the proof of Fermat’s Last you go to a 2-hour movie starting at 11 o’clock, you will What is certain is that 1,500 years later the Greeks Theorem. Together with Fred Diamond, Brian Con - get out at 1 o’clock. This illustrates the following equal - knew the algorithm to generate all whole number solu - rad, and Christophe Breuil, he resolved completely the ity in arithmetic modulo 12: tions to this equation. We know this because Euclid Shimura-Taniyama-Weil Conjecture in the theory of explained the method in Book X of his famous Elements . elliptic . With Michael Harris, he proved the 11 + 2 ≡ 1 mod 12. In modern algebraic notation, Euclid proves that every local Langlands conjecture. More recently, Taylor whole number solution to X 2 + Y 2 = Z 2 has the form established the Sato-Tate Conjecture, another long - This may seem a rather trite variant on our usual standing problem in the theory of elliptic curves. arithmetic, and the reader could legitimately wonder if X = ( a 2 – b 2)c/2 Y = abc Z = ( a 2 + b 2)c/2

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The only way we can have 1796 of his celebrated law of , which (X,Y) ≡ (0,0), (0,4), (1,0), (1,4) mod 5. had previously been conjectured by and X 2 + Y 2 ≡ 0 mod 3 Joseph Lagrange. This was supposed to have been Within less than three years, Yutaka Taniyama and ’s favorite theorem, and he kept coming back to it Goro Shimura (former IAS Member) proposed a daring is to have X 2 ≡ Y 2 ≡ 0 mod 3. This means that 3 during his life, giving eight different proofs. It states that generalization of Eichler’s reciprocity law to all cubic divides X and Y; so that 9 divides X 2 + Y 2 = 3 Z 2 ; so equations in two variables. A decade later, André Weil that 3 divides Z 2 ; so that 3 also divides Z. This is impos - if p is a , then the number of (former IAS Professor) added precision to this conjec - sible, because we had arranged that no whole number square roots of an n in arithmetic ture, and found strong heuristic evidence supporting the greater than 1 divided each of X, Y, and Z. As we have modulo p depends only on p modulo 4n . Shimura-Taniyama reciprocity law. This conjecture reached a contradiction, the only possibility is that our completely changed the development of . initial assumption was flawed, i.e., there could not have On the face of it, this may not seem surprising, but I In the mid-1980s, Gerhard Frey, Jean-Pierre Serre been a solution to would stress that there is no apparent reason why trying (former IAS Member), and Kenneth Ribet (former IAS to solve the equation Member) showed that the Shimura-Taniyama reciproci - X 2 + Y 2 = 3 Z 2, ty law, if true, would imply Fermat’s Last Theorem. X 2 ≡ n mod p Motivated by this, in 1995, Andrew Wiles (former IAS with X, Y, and Z non-zero whole numbers. Member), partly in collaboration with the author, estab - This sort of argument works not only for this particu - should have anything to do with p modulo 4n. Thirty lished many cases of the Shimura-Taniyama reciprocity lar equation. A beautiful theorem of Hermann years after I first learnt how to prove this theorem, it still law and hence finally proved Fermat’s Last Theorem. Minkowski (1890) and Helmut Hasse (1924) says that if seems miraculous to me. Meanwhile, in the mid-1970s, Robert Langlands is any homogeneous quadratic polynomi - Gauss’s theorem also provides a very effective way of (Professor Emeritus, School of Mathematics) had the Q(X1 , ..., Xd ) al in any number of variables with whole number coeffi - determining the number of square roots a whole number extraordinary insight that the ideas of Eichler, Taniya - cients, then has in arithmetic modulo a prime number p. For exam - ma, and Shimura were a small part of a much bigger pic - ple, one could ask how many square roots 3 has in arith - ture. He was able to conjecture the ultimate reciprocity

Q(X 1, ..., X d ) = 0 metic modulo 20132011, which is a prime number. You law, an enormous generalization of what had gone could, in theory, check all the 20132011 possibilities and before, which applies to any number of equations, of any has a non-zero solution in whole numbers if and only if it determine the answer, but (without a computer) this degree in any number of variables. In the last ten years, has a non-zero solution in all (real) numbers and a prim - would take a very long time. On the other hand using the ideas introduced by Wiles, there has been itive solution modulo m for all positive whole numbers much progress made on Langlands’s reciprocity conjec - . (We call a primitive solution modulo if ture, but much more still remains to be done. m (X1, ..., Xd ) m 20132011 = 1677667 × 12 + 7 One striking feature of all the non-abelian reciproci -

Q(X 1 , ... , Xd ) ≡ 0 mod m, so that 3 has the same number of square roots in arith - ty laws is that the formula for the number of solutions is metic modulo 20132011 as it does in arithmetic modulo given in terms of of certain curved

but no integer greater than 1 divides all the Xi .) This is 7. But it is very quick to list the squares modulo 7: —an extraordinary connection between solving actually a very practical criterion. It may appear that one algebraic equations and geometric . In the case needs to check for solutions to our equation in arith - 02 ≡ 0, 1 2 ≡ 1, 2 2 ≡ 4, 3 2 ≡ 2, 4 2 ≡ 2, 5 2 ≡ 4, of the Shimura-Taniyama reciprocity law, the relevant metic modulo m for infinitely many m. However, one 62 ≡ 1 mod 7. symmetries are those of the “hyperbolic plane.” The can find a single integer m 0 (which depends on the hyperbolic plane can be thought of as a circular disc ) with the property that, if Thus 3 has no square root in arithmetic modulo 7 and so (without its boundary), but with an unusual notion of Q Q(X1 , ..., Xd ) = 0 has a primitive solution modulo m 0, then it also has by Gauss’s theorem it has no square root in arithmetic . For two points near the center of the disc, their a primitive solution modulo m for any other positive modulo 20132011. (A good thing we didn’t waste our “hyperbolic” distance is similar to their usual distance, whole number m. time checking all 20132011 possibilities!) but are increasingly distorted near the edge of However, for higher degree equations, the correspon - One could ask for a similar method that given any the disc. The hyperbolic plane and its symmetries were ding theorem can fail. For instance number of in any number of variables helps illustrated in some of Escher’s woodcuts, like the one one to determine the number of solutions to those equa - below. In the hyperbolic world, all the fish in Escher’s 3X 3 + 4 Y 3 + 5 Z 3 = 0 tions in arithmetic modulo a variable prime number p. print are to be thought of as having the same size. Such results are referred to as “reciprocity laws.” In the M O C has non-zero solutions modulo every positive whole 1920s, Emil Artin gave what was then thought to be the . R E H C

number (and it has a solution in the real numbers), but most general reciprocity law possible—his abelian reci - S E C M .

it has no non-zero solution in whole numbers. (This procity law. However, Artin’s reciprocity still only W W W .

famous example was found by Ernst Selmer, former IAS applied to very special equations—equations with only D E V R E

Member.) Nevertheless, when studying the whole num - one variable that have “abelian Galois group.” S E R S

ber solutions to any polynomial equation, the study of Stunningly, in 1954, Martin Eichler (former IAS T H G I R

solutions modulo is often a key tool. Member) found a totally new reciprocity law, not L m L A . D

______included in Artin’s theorem. (Such reciprocity laws are N A L L

often referred to as non-abelian.) More specifically, he O H - Y N

ore than 1,800 years ago the Chinese text Sun Zi found a reciprocality law for the two variable equation A P M O Suan Jing contained a statement of what is now re - C R M E 2 3 2 H C

ferred to as the Chinese Remainder Theorem. This the - S Y + Y = X – X . E . C .

orem gives a very efficient algorithm that reduces the M E H T 2

study of the solutions to a polynomial equation in arith - He showed that the number of solutions to this equation 1 0 2 metic modulo a whole number m, to the study of the in arithmetic modulo a prime number p differs from p by © same equation in arithmetic modulo the factors of m of the coefficient of q p in the formal (infinite) product Circle Limit III M. C. Escher the form pa, where p is a prime number and a is a positive ______whole number. In fact, it turns out that the key case to q(1 – q)2 (1 – q 11 )2 (1 – q 2)2 (1 – q 22 )2 (1 – q 3)2 consider is when m is a prime number. Thus, for the rest (1 – q 33 )2 (1 – q 4)2 ... = q – 2 q 2 – q 3 + 2 q 4 + q 5 will conclude by discussing one further question about of this article, we will only consider arithmetic modulo a modular arithmetic which has seen recent progress. + 2 q 6 – 2 q 7 – 2 q 9 – 2 q 10 + q 11 – 2 q 12 + . . . I prime number p. Instead of asking for a rule to predict how many solu - Recall that a prime number is a whole number greater tions an equation will have in arithmetic modulo a vary - than 1, which is only divisible by 1 and by itself. Exam - For example, you see that the coefficient of q 5 is 1, so ing prime p, one can ask about the statistical behavior of ples are 2, 3, 5, 7, 11, 13, 17, and 19, but not for instance Eichler’s theorem tells us that the number of solutions as the prime varies. Going back 15, which is divisible by 3 and 5. Every positive whole to the simple case of a quadratic equation in one vari - number can be written uniquely as a product of prime Y 2 + Y = X 3 − X 2 able, Lejeune Dirichlet showed in 1837 that for a fixed numbers (up to order). In some way, prime numbers are whole number m, which is not a prefect square, the a bit like the atoms of which all other whole numbers are should have 5 − 1 = 4 solutions in arithmetic modulo 5. equation composed. You can check this by checking the twenty-five possibili - X 2 ≡ m mod p The first really great achievement in the study of ties for (X,Y ) modulo 5, and indeed you will find exactly modular arithmetic was Carl Friedrich Gauss’s proof in four solutions: (Continued on page 8)

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MODULAR ARITHMETIC (Continued from page 7)

has two solutions for half of all prime numbers p and no number of variables of any degree. These celebrated con - solutions for half of all prime numbers p. This may seem jectures led to a revolution in arithmetic algebraic geom - a natural answer, but Dirichlet’s proof was very subtle, etry. Weil’s conjectures were finally proved by Pierre combining Gauss’s reciprocity law with ideas from com - Deligne (Professor Emeritus, School of Mathematics) in plex analysis. In 1880, Ferdinand Frobenius generalized 1974. Dirichlet’s theorem to any equation in one variable. For Returning to the equation other equations, the correct answer may be harder to guess. For instance, the equation Y 2 + Y = X 3  X 2,

X 4 ≡ 2 mod p Hasse’s theorem tells us that asking for what fraction of primes this equation has—say, ten solutions in arith - has no solution for 5/8 of all prime numbers p; has two metic modulo p—is not interesting; the answer will solutions for 1/4 of all prime numbers p; and has four always be 0. Rather the natural question is to consider solutions for 1/8 of all prime numbers p. the normalized error term Sato-Tate Distribution for Δ and p <1,000,000 What about such density theorems for equations in (drawn by William Stein) more than one variable, like Np p — The Sato-Tate density theorem has recently been proven √p. Y 2 + Y = X 3  X 2 ? (by Laurent Clozel and Michael Harris, both former IAS By Hasse’s theorem, this will be a (real) number lying Members; Nicholas Shepherd-Barron; and the author), In this case, Hasse showed in 1933 that the number of between 2 and 2, and one can ask how it is distributed not just for this equation, but for all cubic equations in solutions in arithmetic modulo p, which we will denote in this interval. Is the error often as large as Hasse’s the - two variables. The proof combines the arguments of , is of the same order of magnitude as . More precise - orem allows, or is it usually smaller and only rarely at the Dirichlet and Frobenius with an infinite series of new Np p ly, he showed that differed from by at most . He extreme? In 1963, Mikio Sato and John Tate (both for - cases of Langlands’s reciprocity law. There should, of Np p 2√p proved this for all cubic equations in two variables. mer IAS Members) independently conjectured the cor - course, be density theorems for any number of equations In 1949, Weil conjectured an enormous generaliza - rect density theorem—the error should be distributed in any number of variables of any degree, but these remain — tion of Hasse’s bound to any number of equations in any like (1/2π) √4  t 2, a “squashed semi-circle.” very much conjectural. The story is continuing . .. I

HOW OUR BRAINS OPERATE (Continued from page 1) detailed properties. Individuality: all humans are similar, but the The existence of memory. The The pressure-sensing nerve cells in your feet were sending sig - operation of each brain is critically dependent on its individual nals to your brain both before and after you heard that question, details. Your particular pattern of connections between nerve sense of time. Thinking. Human but somehow hearing that question switched your state of con - cells contains your personality, your language skills, your knowl - language. Consciousness. These scious awareness of your feet. There are no useful scientific edge of family, your college education, and your golf swing. descriptions of how this takes place, or of the neural activity We will never know the connectivity pattern of any indi - properties are all collective, high- that this state of awareness of my feet implies. The problem is vidual human. But even without that, there are general ques - level descriptions of events that are that consciousness is an internal personal feeling. I can describe tions essential to our humanness that can be addressed on the implicitly caused by the activity of my conscious awareness of the world to you by using language, basis of the many neurobiological facts now available. but you have no scientific way of verifying that description. The existence of memory. The sense of time. Thinking. huge numbers of interacting nerve I have asked many brilliant scientists about consciousness and Human language. Consciousness. These properties are all collec - cells. Tornados are a high-level have heard an amazing diversity of answers. The artificial intelli - tive, high-level descriptions of events that are implicitly caused description of the motions of gence pioneer Marvin Minsky remarked that the really difficult by the activity of huge numbers of interacting nerve cells. Torna - things that a brain does are all done nonconsciously—tasks like dos are a high-level description of the motions of enormous num - enormous numbers of interacting perceiving the visual world as made up of separate objects. Min - bers of interacting molecules. We want to understand how mind molecules. We want to understand sky further pointed out experiments in which a decision that you emerges from brain, just as we understand how tornados emerge how mind emerges from brain, just declare you have made consciously had in fact been made and from molecules. I will say a few words about the easiest and about begun to be acted on before you were consciously aware of that the deepest items on the list: memory and consciousness. as we understand how tornados decision. In Minsky’s view, consciousness may be a feeble attempt We have associative memory. If you went to a party last Sat - emerge from molecules. of the human brain (a special kind of computer) to describe its urday, the location of the party, whom you spoke with, what own actions in terms of the behavior of an agent called “me.” the meal was, the conversations you had are all linked together. Let’s say someone asks I asked Francis Crick for his view of consciousness. He replied with a long monologue you, “Have you seen Pat since last summer?” You might reply, “He was at a fabulous party about the gene, beginning with Gregor Mendel and his discovery of the discrete nature last Saturday, where I heard that . ..” and on. Items in your memory are linked together of inheritance. It took a couple decades for Mendel’s discovery to become understood as in a rich web, and reminding you of something immediately connects you with items that the central issue of biology. Slowly, the concept of a gene was developed. That concept are associated . Recall is summoned up by association. By began with no plausible basis in the physical or chemical world. It was more than a cen - contrast, all digital computer memory is by address. You tury between Mendel and the ultimate understanding of the basis of inheritance through store information at a location and can retrieve it only knowing the genetic meaning of a DNA by giving the location at which it is stored. molecule. Through this century, biologists John Hopfield is the Martin A. and There is a simple mathematical model describing wrestled with a concept of a gene because Helen Chooljian Visiting Professor in how a particular kind of nerve circuit will behave as an they understood that inheritance stood at Biology in the School of Natural Sci - associative memory. The question of whether this the heart of biology. Crick concluded this ences and the Howard A. Prior Pro - model in fact describes how we have associative mem - lecture (to me) with the simple statement fessor in the Life Sciences, Emeritus, ories then becomes an experimental question of “To me, consciousness is like that—ill- and Professor of Molecular Biology, whether the part of the brain that is responsible for formulated, but lying at the center of what it Emeritus, at Princeton University, associative memory has the expected circuitry and uses is to be human.” where he joined the faculty as a profes - it in the expected fashion. There is increasing evidence I myself would not go quite that far. But I sor of physics in 1964. Hopfield’s re - in the rat, one of our very distant cousins, that a region do share with Crick the belief that we will search has demonstrated how modeling of the brain called the hippocampus has the expected not understand the essence of what it is to be biological processes in the brain can be properties, and has activity patterns consistent with human until we have a viable theoretical used to solve complex computational associative memory. What about in humans, for our understanding of how the human brain func - problems. The beginning of the modern brains contain a hippocampus that is very similar to tions. It may take another century to pro - era of neural networks can be traced to that of a rat? For humans the situation is more complex, duce. Present experimental facts are his pioneering work in the early 1980s. and the current popular hypotheses are more a “Just So” inevitably inadequate. But, like the genetic His current research examines issues story than a viable theoretical description. pioneers, we must try, for how the human Neural circuitry of the rodent such as “thinking” and “perception” in Finally, on to consciousness. Let me ask each of you, brain works is the most fundamental scientif - hippocampus drawn by Spanish the intersection between these two ideas. scientist Ramón y Cajal in 1911 “Are you conscious of your shoes pressing on your feet?” ic question that our humanity poses to us. I

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Gathering Arctic Voices at the Institute, with the Woods to Inspire

may not have the whole answer, I’ve been putting to - BY SUBHANKAR BANERJEE gether bits and pieces in response to that question. Indeed, around the world, the Arctic is thought to be rior to arriving at the Institute as a Director’s Visitor last a remote place disconnected from our daily lives. On the Pfall, I had heard about the Institute Woods, of which contrary, hundreds of millions of birds migrate to the Arc - George Kennan had said: tic each spring from every corner of the earth—including I have lived in the proximity of these Woods for over half Yellow Wagtail from Kolkata—for nesting and rearing a century. They are a friend, a source of inspiration and their young, and resting—a planetary celebration of glob - restoration, and were they to disappear it would be like the al interconnectedness. On the other hand, caribou, disappearance of an old, beloved, and respected friend. whale, and fish migrate hundreds and sometimes thou - sands of miles, connecting numerous indigenous commu - I needed a source of inspiration, as I was editing an nities through subsistence food harvests—local and anthology Arctic Voices: Resistance at the Tipping Point at the regional interconnectedness. However, deadly industrial time. So, I made a request of Peter Goddard, then Director toxins migrate to the Arctic from every part of our plan - of the Institute, and Yve-Alain Bois, Professor in the et, making animals and humans of the Arctic among the School of Historical Studies, who had facilitated my visit: most contaminated inhabitants of the earth. The breast If I am allowed, I would like to make my office in the milk of high Arctic women in some parts of Greenland IAS Woods rather than inside an office building where and northern Canada is scientifically regarded as being as I might be given an office space. Each morning pre- toxic as hazardous waste—a planetary tragedy of global dawn I would like to walk for three hours in the Woods, interconnectedness. Marla Cone tells this tragic story in and same around dusk, making it a full day of work. this volume—she calls it “Arctic Paradox.” Dr. Goddard responded: As you can see, the Arctic is far from being a remote place disconnected from our daily lives. Instead, we’re all “It is good to hear that you are excited about the woods. connected to the northern landscape. In this volume, we You are of course free to walk there whenever you wish, tell many stories of local, regional, and global intercon - but we will still allocate you a normal office!” nectedness—both celebratory and tragic. As it turned out, I really needed a “normal office” — ______where I worked long hours each day to complete my work on the Arctic anthology, a compilation of thirty-nine con - he Arctic, after all, is big—it is the top of our earth, tributions by indigenous activists, writers, conservationists, Tthe ice cap, some call it, but it is so much more, and Caribou Migration I , from the series Oil and the Caribou it’s that so-much-more that this book is about. scientists, and humanities scholars, as well as photographs (Arctic National Wildlife Refuge), 2002 and drawings by sixteen artists. What follows is an excerpt The Arctic has become our planet’s tipping point— from my introduction in Arctic Voices . climate change is wreaking havoc up there. Resource ______not normal, I was told, but no one in town said the Arctic wars continue to spread. Industrial toxins continue to was getting warmer. I now read that the bears of Hudson accumulate widely. But also, the voices of resistance are ow do we talk about the Arctic? Bay will disappear within a few decades at best, or within gathering, are getting louder and louder—and that is the H How do we think about the Arctic? a decade at worst, because these days ice is forming later in story this volume presents. It is the noise and the music How do we relate to the Arctic? autumn and melting sooner in spring, leaving the bears of all our voices bundled together. And, why talk about the Arctic, now? These are some longer on land, where they must wait and starve. This grue - Arctic Voices doesn’t have a linear structure; it isn’t questions we explore, through stories, in this volume. some photograph of death produced in me a desire to live arranged chronologically or even geographically, but Along the way, we talk about big animals, big migra - in the wild, with the polar bears. rather as a web of interconnections with loosely defined tions, big hunting, big land, big rivers, big ocean, and big ______themes that you may read in any order you wish. I have sky; and also about big coal, big oil, big warming, big spills, found plenty of things in common between essays—for big pollution, big legislation, and big lawsuits. use photography to raise awareness about the Arctic, example, the spectacled eiders that winter in the frozen And we talk about small things, too—small animals, Ibut I never would have imagined that my photographs Bering Sea, written about by Nancy Lord, also nest in the small migrations, small hunting, small rivers, small warming, would be used on the U.S. Senate floor to argue against oil Teshekpuk Lake Wetland that Jeff Fair writes about; both small spills, small pollution, small legislation, and small drilling in the Arctic National Wildlife Refuge—yet that writer Velma Wallis and artist Annie Pootoogook use lawsuits. is exactly what Senator Barbara Boxer did and won a cru - stories and art as an outlet for healing as they both ______cial vote on March 19, 2003. Nor did I imagine that my address alcoholism in their unique ways; and common exhibition at the Smithsonian Institution would be cen - words take on new meaning, for example, Seth Kantner n the Arctic, impacts of climate change can be seen sored and become the topic of a Senate hearing at which and Matthew put the word subsistence on its Iand/or experienced everywhere. Indeed, the Arctic is Senator Richard Durbin would support my work, or that head, while Andri Snaer Magnason tells us how Alcoa warming at a rate double that of the rest of the planet. We later a Senate investigation would follow. But when then- hijacked the word sustainability in Iceland and Green - tell many stories of climate change in Arctic Voices. Senator Ted Stevens during a May 2003 Senate debate land. I’m sure you will find more such interconnected - At the same time, I am realizing that there is an Arctic said that President Jimmy Carter and I were giving “mis - ness, and I surmise that you will begin to think and talk paradox: that oil, coal, and gas, the burning of which has information to the American public”—effectively calling about the Arctic differently than you did before. And caused unprecedented Arctic warming, are the same non - us liars—then I did fear possible deportation, and realized perhaps you’ll find an answer to the question, “Why renewable resources whose extraction projects are expand - that if I were to have a voice in conservation in the U.S., should I care about the Arctic?” I ing rapidly in the Arctic—terrestrial and offshore. I must become a U.S. citizen. So I did. Excerpted from Arctic Voices: Resistance at the Tipping There are resource wars—for oil, gas, coal, and miner - Secretary of the Interior Gale A. Norton, during a Point edited by Subhankar Banerjee © 2012, reprinted by als—everywhere in the Arctic—from Alaska to Siberia, March 12, 2003, congressional testimony, famously arrangement with Seven Stories Press, New York. with Nunavut and Greenland along the way. In Arctic described the Arctic Refuge coastal plain as an object of Alaska, these wars have intensified since I first arrived conceptual art—“a flat white nothingness.” … Then, on there more than a decade ago. November 5, 2005, Senator Stevens said on PBS Subhankar Banerjee, a Director’s Visitor in fall 2011, ______NewsHour with Jim Lehrer, “This is the area in wintertime. is an Indian-born American photographer, writer, and And I defy anyone to say that that is a beautiful place that activist. Over the past decade, he has worked for the ou might wonder how someone with an Indian-sound - has to be preserved for the future. It is a barren wasteland, conservation of ecoculturally significant areas of the Ying name like mine, someone from the south, comes frozen wasteland.” Arctic and to raise awareness about indigenous human to concern himself with all things northern. Here is how it Arctic Voices paints a very different picture—we present rights and climate change. He is the author of Arctic all began. In 2000, I left my career as a scientist and was the Arctic neither as a frozen wasteland nor as a pristine National Wildlife Refuge: Seasons of Life and wandering aimlessly from Florida to British Columbia look - wilderness, but, instead, simply as home for numerous Land (Mountaineers Books, 2003), and his photo - ing for inspiration for a photography project; I had found species—animal and human—who either visit for a while graphs have been exhibited in more than fifty muse - none when, in late October, I arrived with two friends in or live there year-round. ums. He will read from Arctic Voices (Seven Stories Churchill in subarctic Canada—a popular tourist destina - ______Press, 2012) at Labyrinth Books in Princeton on tion. There, polar bears gather along the Hudson Bay and October 9. He received a 2012 Cultural Freedom wait on land for the bay to freeze over. Once on ice, they ver the past decade, many people have asked me, Award from the Lannan Foundation. hunt and eat. I took a photo of one bear eating another— O“Why should I care about the Arctic?” While I still 9 G12-12800_IAS_SUMMER12.qxp 8/16/12 2:02 PM Page 10

Liberal Democratic Legacies in Modern Egypt: The Role of the Intellectuals, 190 0–1950

rent within the intellectual community. In this article, I salons, and the theater and cinema industries. Old and BY ISRAEL GERSHONI will first describe the salient features of this community young, men and women, Christians, Jews and Muslims, and the contexts in which they operated, and then focus and elite and non-elite strata participated in this inclu - reedom is the ultimate virtue of mankind”; on the liberal modes of thought developed by two of the sive civil activity within public spheres. However, the “F“ is the only political system of mod - most representative intellectuals. This intellectual cur - intellectuals constituted the hardcore center of this effen - ern man and modern society”; “Therefore, Egypt must be rent coalesced and exerted its influence during the inter - di liberal public sphere: they were the idea makers, the committed to freedom and democracy.” These are the war era (1919–39) until the mid-1950s, galvanizing and cultural producers, the writers and artists. words of ‘Abbas Mahmud al-‘Aqqad in his book Hitlar fi Historically, intellectual efforts to liberalize and al-Mizan (Hitler in the Balance ), which aroused sharp pub - The effendiyya , groups of urban democratize Egyptian life were underway as early as the lic interest in Egypt and the Arab world when it was pub - educated middle classes that emerged first decade of the twentieth century. In particular, the lished in Cairo in early June 1940. The book was written thought of Ahmad Lutfi al-Sayyid (1872–1963), his jour - when Hitler was at the height of his military successes, and expanded in the first half of the nal al-Jarida, and Hizb al-Umma (the “Party of the and it was widely assumed that nothing would thwart his twentieth century, provided the fertile Nation”) laid the groundwork for the emergence of a lib - advances. ‘Aqqad’s book leveled a harsh attack on Hitler social grounds for the emergence of eral democratic worldview from 1907–15. True, this was a and Nazism. Through his analysis of Hitler’s complex and patronizing, elitist concept of a democracy that assumed deranged personality, ‘Aqqad deconstructed Nazi racism, liberal, multivocal, and heterogeneous the intellectuals would play the Platonian role of philoso - dictatorship, and imperialism. He portrayed Hitler and public spheres. They promoted and pher-kings whose mission was to guide and control the Nazism as the ultimate danger not only for freedom and maintained freedom of the press, which democratic process. Because of the masses’ ignorance and democracy, but also for modernity, the very existence of irrationality, they were not supposed to take part in the modern man and enlightened culture. In ‘Aqqad’s view, encouraged and expanded public liberal democratic milieu. Most of the liberal intellectual the merits of a liberal democracy were rooted in: individ - discourses both in Egypt and leaders whom I will discuss started their careers as disci - ual freedoms and , constitutionalism, a par - throughout the Arab world, and were ples of Lutfi al-Sayyid and the al-Jarida school. With time, liamentary and multiparty system, the separation of they would go on to challenge this elitist idea of a democ - powers, equality for all citizens, cultural pluralism, and responsible for the flourishing of a racy and champion a more inclusive notion of a liberal the unquestionable legitimacy of political opposition. pluralistic culture of everyday life. democracy. This intellectual activity reached its zenith When ‘Aqqad (1889–1964) expressed these views in during the interwar era when its social, political, and cul - the early years of the Second World War, his liberal dem - institutionalizing a strong tradition of liberal democratic tural impact was most discernible. ocratic worldview had fully coalesced. Already in his thought in Egypt and in the Arab Middle East. The How did these intellectuals conceive of a liberal early fifties, he was an established and well-known intel - intellectual community was active in a relatively sympa - democracy? Generally, one can say that for them, a lib - lectual active for more than three decades. In hundreds thetic and friendly environment underpinned by two eral democracy was an organizing form of society and of articles published in the Egyptian press, and particu - essential elements: the first was the very existence of polity whose declared goal was to establish a liberal and larly in his book The Absolute Rule of the 20th Century (al- parliamentarism; the second was the emergence and just order that would guarantee the free will of all of its development of a strong civil society. citizens, while defending civil rights and liberties. A lib - From 1923 to 1952, parliamentary government eral democratic government was to mediate between served as the basic framework within which Egyptian the wills of individuals and the collective interests of political, social, and cultural life evolved. Following the society through elections, representation, and majority- Great War, the eruption of the anti-colonial revolution based decision-making that protects the minority. Its of 1919 forced British authorities to grant Egypt formal aim was to facilitate the institutionalization of courts independence in February 1922. Egyptian independence and agencies, which would arbitrate conflicting inter - facilitated the promulgation of a liberal constitution in ests. This democracy would not pose any limitations on 1923 that called for a two-house Parliament, Chamber individual freedoms except when the individual’s deed of Deputies, and Senate, and the immediate establish - infringed upon the rights of others, the collective, or ment of a parliamentary monarchy headed by King the system itself. Fu’ad. For the ensuing thirty years, Egyptian political life This concept of a liberal democracy originated in consisted of a parliamentary system in which political three major sources. The first was the Mediterranean parties competed for office in periodic national elec - Greco-Roman legacy, particularly Plato and Aristotle, tions. This system endured until the July 1952 Revolu - and the Roman republican constitutionalist tradition as tion and the declaration of the Republic of Egypt in expressed by Polybius and Cicero. The second major 1953. It dismantled the autocracy of the Khedival rule, source was the British political tradition of the seven - eroded the authoritarian political culture of the late teenth through nineteenth centuries as well as the polit - Egyptian-Ottoman oligarchy, and weakened British ical thought from the French Enlightenment of the colonial rule. It encouraged ethnic pluralism and reli - eighteenth century. Egyptian intellectuals demonstrated gious tolerance, reduced the presence of the police and intimate familiarity with the Greco-Roman intellectual army, and cultivated rich cultural activity with minimal tradition, part of which was translated into Arabic first state intervention. in the Middle Ages, and later in the modern era. They The relative success of Egyptian parliamentary gov - also read Hobbes, Locke, , , ernment was based on a mature civil society that devel - Rousseau, and the nineteenth-century writings of Ben - oped distinct liberal public spheres. It emerged in the late tham, Mill, and Tocqueville. As activists, they tended to nineteenth century, reached maturity after the Great select the liberal themes, principles, and practices that War, and flourished during the interwar era. This civil were most easily translatable to their society’s needs and Ruz al-Yusuf reacting to the outbreak of the Second World War. society gave birth to a liberal public sphere, one that I conditions. They were dedicated to comprehending and “History repeats itself: The end of Hitler at the hands of democ - define as the “ effendi liberal public sphere.” Indeed, this adapting these bodies of thought to their local political racy.” Britain, France, and Egypt portrayed as Allied Forces civil society originated in an effendi social milieu. The culture and to disseminating liberal democratic princi - against Nazi Germany. September 9, 1939. effendiyya , groups of urban educated middle classes that ples and practices among broader sectors of society in emerged and expanded in the first half of the twentieth order to create a solid tradition of an Egyptian-Arab lib - Hukm al-Mutlaq fi al-Qarn al-‘Ishrin ), published in 1929, century, provided the fertile social grounds for the emer - eral democracy. The third and perhaps secondary source ‘Aqqad reaffirmed his commitment to democracy and his gence of liberal, multivocal, and heterogeneous public from which these intellectuals drew inspiration was rejection of any form of absolutism, oligarchy, aristocra - spheres. First, the effendiyya promoted and maintained local. They extracted insights and theories from classical cy, and autocratic monarchial rule, and in particular Fas - freedom of the press, which facilitated the production of cism, Nazism, and, in a different way, Communism. As a a diverse variety of hundreds of dailies, weeklies, and Israel Gershoni, a Member in the School of Historical representative of the Wafd party in the Egyptian parlia - monthlies, and other print media products. Second, this Studies (2011 –12), is teaching the history of the mod - ment, and later as the intellectual leader of the Sa‘adist explosion of print culture encouraged and expanded pub - ern Middle East at Tel Aviv University. His recent Party and its representative in the Chamber of Deputies, lic discourses both in Egypt and throughout the Arab book, coauthored with James Jankowski, Confronting ‘Aqqad was one of the most consistently democratic world. Third, the effendiyya was also responsible for the Fascism in Egypt: Dictatorship versus Democracy in activists in Egyptian politics and culture. pluralistic culture of everyday life that flourished in cof - the 1930s , was published by Stanford University Press However, ‘Aqqad was by no means exceptional. His fee and teahouses, civil associations, entrepreneurial in 2010. ideas and activities aptly reflected the mainstream cur - projects, political organizations, social clubs, cultural

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Islamic political philosophers, particularly, al-Farabi, Ibn Islamic orthodoxy; he sought to neutralize its power and Mustaqbal al-Thaqafa, he explicitly excluded language Sina, Ibn Rushd, and Ibn Khaldun. authority in society, politics, and culture. Within his and religion from his conception of Egyptian national Despite their reliance on classical texts, Egyptian intel - struggle, he attempted to subordinate al-Azhar, the tra - identity. As such, Muslims, Christians, Jews, and other lectuals emphasized that an Egyptian liberal democracy ditional Muslim institution of academic learning, to the ethnic or linguistic communities were all equally Egyp - was principally a modern concept. From their perspective, Ministry of Education and to reduce its sway over public tians. Husayn’s liberal, inclusive approach to Egyptian it was an integral part of the formation of a modern secu - education. He was also a republican by nature and strove collective identity assumed that all Egyptians were equal lar worldview and way of life. Therefore, their liberal to undermine the authority of the monarch, which with citizens who should elect their representatives through democratic weltanshauung was formulated as an antagonist al-Azhar’s support, tried to establish an autocratic form the institutions and mechanisms of multiparty parlia - to institutionalized religion, Khedival autocracy, authori - of government. On a different level, in his quest for mentary government. tarian Ottoman political norms, and the Egyptian- democracy, Husayn turned his sights against Fascism. He Ottoman aristocracy and oligarchy. This democratic viewed it as a dictatorial system that silenced freedom of liberalism was inextricably linked to , the pri - thought, suppressed civil rights, and transformed human macy of human reason, science, progress, modernism, sec - beings into mere atoms within state machinery, thereby ularism, , and a strain of separation between imprisoning them. In taking man’s freedom, Fascism— religion and the nation-state. As Albert Hourani by its very nature—deprived man of his humanity. described their worldviews in his classical book Arabic However, Husayn understood that there was a built- Thought in the Liberal Age, 1789 –1939 (1962), “according in structural problem in his society: a majority illiterate to such thinkers, human society is standing, by the irre - populace in a liberal democracy. He never ignored the versible and irresistible natural law of progress, towards an divide between the thin layer of the educated elite and ideal state, of which the marks will be the domination of the broader illiterate sectors of his society. Disturbed by reason, the extension of individual liberty ... and the the structural problem of illiteracy, Husayn dedicated his replacement of relations based on custom and status by life and career to education, which he regarded as the those based on free contract and individual interest.” most effective agent of democratization and liberaliza - Prominent within this intellectual community of dis - tion of Egyptian life. Following Plato, Rousseau, and course were about thirty luminary liberal intellectuals. Mill, he held that through education and literacy, one Some of them were more dominant: ‘Abbas Mahmud al- could enlighten and refine the broader sectors of society ‘Aqqad, Ahmad Amin, Salama Musa, Taufiq al-Hakim, and include them in civil democratic politics. He ‘Ali ‘Abd al-Raziq, and ‘Abd al-Razzaq al-Sanhuri. The espoused the notion that education should be developed intellectual voices of women were equally important, and promoted by the government, which should regard particularly those of Huda Sha‘arawi, Nabawiyya Musa, the education of all citizens as its ultimate priority. He and Labiba Ahmad. All of these intellectuals were of the worked for the spread of education through the Egyptian same generation. They were born at the end of the nine - University and his governmental service, which culmi - teenth century and were active as public intellectuals, nated in his appointment as Minister of Education in the particularly in the interwar era. Two of these, however, last Wafd government (1950 –1952). Without proper were most exceptional: Taha Husayn (1889–1973) and education, Husayn held that one cannot comprehend Al-Ithnayn wa al-Dunya responding to Fascist Italy’s brutal Muhammad Husayn Haykal (1888–1956). The Egypt - the essence of democracy, its ideas, institutions, and its conquest of Albania during spring 1939. “Her guardian . .. ian-Arab public considered these two individuals to be modus operandi. [her robber]. Mussolini: Come my dear, I will hug you because representative thinkers of the day and the most authori - In his most important book, Mustaqbal al-Thaqafa fi I want you. Albania: Please, loosen your grip, I am dying.” tative proponents of a liberal democracy. Misr (The Future of Culture/Education in Egypt ), which was April 17, 1939 Taha Husayn—referred to as the “Doyen of Arab published in 1938, Husayn discussed the inextricable Culture”—was revered as the greatest thinker of this bond between democracy and education. He argued that Muhammad Husayn Haykal, an equally influential generation. He was the most influential promoter of a education for democracy must begin in elementary school contemporary of Husayn’s, was committed to similar lib - liberal democracy in Egypt and the Arab world. His and should be the highest priority of “the future of culture eral principles. Haykal was more a liberal than he was a impact on larger educated publics was unprecedented. in Egypt.” Borrowing from Rousseau, he posited that men democrat. If Husayn was a republican, then Haykal was a As early as the 1920s, Husayn developed a solid liberal are naturally free. He wrote: “One who gives up his free - monarchist. Whereas Husayn appealed to the entire democratic worldview, which was rooted first in the dom, gives up his humanity.” The negation of freedom body politic, Haykal was primarily concerned with the Greco-Roman tradition, particularly in Aristotelian educated elite. Husayn assumed that liberty and equality thought. However, in contrast to Aristotle, who was hes - Taha Husayn held that through were compatible, while Haykal assumed that they were itant to confer democracy on the whole body of citizens, dramatically opposed and contradictory. Haykal, a con - Husayn’s perspective was all-inclusive. He held that sov - education and literacy, one could servative liberal, regarded freedom as an asset mainly of ereignty emanates from the people and therefore enlighten and refine the broader the enlightened sophisticated elite. If democracy were “democracy is a political system of the people, by the open to all, it would not award sovereignty to the people, people, for the people.” Democracy was always prefer - sectors of society and include them in but to the crowd. The crowd in its nature was illiterate, able to monarchy, aristocracy, or oligarchy, and in mod - civil democratic politics. He espoused irrational, imbued with superstitions, and not in need of ern times, to Fascism, Nazism, and Communism. Husayn the notion that education should be freedom, but stability and security, which only a more was highly conversant with eighteenth-century French authoritarian or autocratic government could provide. Enlightenment thought, especially that of Rousseau. developed and promoted by the Haykal was highly influenced by Plato’s notion of Husayn did his second Ph.D. in the Sorbonne in Paris government, which should regard philosopher-kings and the reworking of this notion set (1913–19) and among his studies was French modern the education of all citizens as its forth by the prominent tenth-century philosopher, al- philosophy. In line with Rousseau’s ideas, he regarded Farabi. Haykal regarded the philosopher-kings of Egypt - society as a community of citizens possessing inalienable ultimate priority. ian society as those refined and talented individuals who natural rights of freedom and justice. To ensure these underwent massive Europeanization and received a rights, Husayn was a proponent of state subordination to contradicts the nature of man. A human being is both an modern education. From a social perspective, he regard - an agreed-upon constitution that protects citizens from individual and a social entity: society must ensure a citi - ed the landed elite as the source for these philosopher- tyranny. Like Rousseau, Husayn’s solution to the inher - zen’s freedom and, in turn, a citizen must contribute to kings. Like Husayn, Haykal challenged the old ent tension between the ruler and the ruled was that society. For Husayn, such a social contract and relations Egyptian-Ottoman oligarchy but sought to replace it “men can be both ruled and free if they rule themselves.” were achievable only in a liberal democracy. with a new aristocracy defined by its virtue and educa - Thus, government would ensure the natural rights to Husayn promoted the idea that only a truly free dem - tion and sheltered by the constitutionalist monarchy. freedom of its citizens, and citizens, in turn, would ocratic citizen can be a loyal patriot. Thus, the purpose Haykal believed that a liberal democratic government demonstrate loyalty to the state and participate in its of education was to instill the notion of proper citizen - should first defend the free thought and activity of this conduct through representative parliamentary institu - ship alongside loyalty to the nation-state, since only educated elite, who would protect freedom and civil tions. All citizens—including women—had the natural educated and free citizens who understand the depths of rights for all and lead and control the processes of mod - right to participate in the political democratic process. their own freedom will be willing to sacrifice themselves ernization and progress. In such a system, freedom of thought, freedom of the for their nation’s freedom. Husayn’s definition of Egypt - Despite his elitist concept, Haykal did not preclude press, public opinion, and cultural pluralism reign. ian nationality was highly liberal and inclusive, extend - open entry into this educated governing elite. Anyone, As part of his promotion of liberal democratic princi - ing to all dwellers of the Nile Valley—irrespective of men and women, who met the criteria of education and ples and values, Husayn waged a scathing attack on their ethnic origin, language, and religion. In his book (Continued on page 15)

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SYMPLECTIC PIECE (Continued from page 1)

chaos in the restricted three-body problem. This problem possible orbits that the satellite can trace out, if given the their own musical material, map out a unique composition describes the movement of a satellite under the gravity of appropriate initial data, allows for many random patterns using this material, and interact with the other performers. two big bodies, say the earth and the moon, in a rotating of transitions between earth and moon. I also found inspiration in John Zorn’s Cobra and coordinates system in which the earth and the moon stay One day at lunch, Derek pressed me on the question of Butch Morris’s early Conductions , works from the 1980s at fixed positions. The chaos in the system is illustrated deterministic chaos. He joked that it could be an ideal start - that present a series of ground rules from which a com - by putting about ten satellites initially at almost the same ing point for a Cage-inspired composition. I pounced on position takes shape. I eschewed the use of Cage’s position with almost the same velocity. the opportunity and suggested that he now had no excuse “chance” in favor of “risk,” since (as Morris writes) When the system starts evolving, the program shows not to accept the commission. Over several more lunches “risk insinuates a certain kind of challenge; chance colorful trackings of the paths of the satellites as they with Derek, he became convinced. We continued to dis - doesn’t necessarily do that.” evolve from a simple single orbit to a complex multicol - cuss the logistics of how to map some of the formal aspects I titled the work Orbit Design, for three or more per - ored tangle of orbits, once the orbits of the different of the three-body problem onto a live musical performance. formers . A set of guidelines assists the performers to satellites start separating. Members Peter Albers and Alvaro Pelayo provided input as structure a path of decision-making as events unfold. Among those in the audience at my 2010 IAS lecture did local and painter Ed Belbruno, whose The piece taxes their skills and perception via materi - was composer Derek Bermel, then the newly appointed ideas for using dynamical systems and chaos theory to al chosen by them. I therefore leave the decisions—the Artist-in-Residence. As I got to know Derek, I realized obtain new low-energy pathways to the moon resulted in “risk”—in the hands of the performers themselves. As that he was interested in mathematics. So when I began the rescue of the Japanese spacecraft Hiten in 1991. Cage wrote, “My work became an exploration of non- planning my upcoming program in symplectic dynamics, “The Symplectic Piece” would be a musical represen - intention . . . making my responsibility that of asking I thought it would be quite something if our resident com - tation of the “gravitational choreography.” A body enters questions instead of making choices.” poser wrote what we would come to refer to as “The Sym - another body’s neighborhood, orbits around it and then In Orbit Design , each performer establishes a unique plectic Piece”—a musical composition inspired by leaves, perhaps visiting another body, returning, and so vocabulary by making a selection from various musical symplectic geometry. on. Working from this idea, Derek set to work developing parameters. Typical parameters might include: pitch (fre - Symplectic geometry is a large field in mathematics, a musical algorithm for three or more players. quency—organized into modes, microtones, chords, and with connections to physics, low-dimensional , But how exactly would Derek do this? This was the big scales, which suggest the notion of octave equivalence , or and geometry, as well as the theory of dynamical systems. question ... other sets); pulse (elemental rhythmic units); note length The modern fields of symplectic geometry and dynamical (which might include articulation ); tempo (speed); volume systems originate in Henri Poincaré’s work on celestial Derek Bermel, Artist-in-Residence, writes: (loudness); color (timbre, including specific effects and mechanics. The term “symplectic” was introduced in nce I decided to take the bait and write “The Sym - strength of timbral modulation); and spatial placement/ 1939 by Hermann Weyl, a Faculty member at IAS, as a Oplectic Piece,” I tried to narrow down my options. My motion (physical location of performers). verbum pro verbo for “complex” in his important work on first questions: How could I create an auditory manifesta - Combining these parameters generates compound the classical groups—the so-called line complex group tion of something as abstract and complicated as symplec - musical events comprising series, sequences, or patterns. became the symplectic group. tic dynamics? What kind of piece would this be? Would it These compound parameters include: density (pulse + Derek and I had several conversations during lunches carry hallmarks of a particular musical style? What instru - tempo); duration (pitch + pulse layer + tempo); melody about the potential for “The Symplectic Piece.” And he ments would play? What would make sense for a tribute to (pitch + pulse); and contour (register + tempo). continued to attend occasional lectures on the subject, John Cage? I was daunted by many issues, not the least of A third, more complex level of relationship that occurs searching for a way to map symplectic geometry onto a which was the specter of serial composition, which had pro - between performers—the overall language of the piece— musical score. But despite many entertaining discussions, duced a good deal of drab music. includes: rhythm (layers of pulse); orchestration (layers of the crucial inspirational idea was missing. With the start One of my problems with serialism is that, though it timbre); harmony/cadence (layers of pitch); and counterpoint of the symplectic dynamics program last fall, “The Sym - aspires to be “mathematical” in its ordering, its underpin - (layers of pitch + pulse). plectic Piece” was still just floating around in the ether. nings nonetheless seem to involve little more than basic My actual “composition” merely supplies instructions Then one day Derek mentioned that he had been invit - arithmetic (pitch sets, inversions, retrogrades, retrograde- as a sort of musical grammar to the performers. Each per - ed by the Look and Listen Festival in New York to compose inversions, etc.) rather than higher-order concepts. I was former must select their own narrow range of values that a commissioned work to mark the one-hundredth birthday of searching for an open approach that was less chained to apply to their own musical parameters. For example, a flute the avant-garde composer John Cage, said to be one of the formal details and more genuinely inspired by symplectic player might choose initial pitch (and register) values by most influential American composers of the twentieth cen - dynamics’ principles. Other (in my opinion) more success - selecting the set of chromatic half steps between c3 and tury. However, besides having too many commitments ful models existed for music based on mathematical f#3 inclusive; for initial pulse and tempo values, the flutist already, Derek felt that a Cage composition might be beyond processes—such as the stochastic music of Iannis Xenakis might select half notes (semiminims) and quarter notes his comfort zone as a composer, so he declined the offer. or the spectral music of Gérard Grisey and Tristan Murail. (crotchets) at sixty beats per minute. The flutist continues But as Cage once said: “Ideas are one thing and what But these were more closely linked to theory to select value ranges for volume, timbre, spatial place - happens is another.” and , respectively, and seemed therefore ment in the room, and any other data relevant to the per - Or, to put it another way, I just had to talk Derek into it. only tangentially related to symplectic systems. formers’ vocabulary, making sure it is clearly notated. Since good preparation is everything, I conducted some Then there was the problem of abstraction. Since music The performers then decide which parameters will research via my subscription to the music service Rhap - is composed of vibrating atoms of air, I needed an analogy remain static (constant) and which ones will expand. For sody, searching for John Cage and selecting “Play All Top that was dynamic. Helmut suggested the title Orbit Design dynamic parameters, the performers indicate how (in Tracks.” Initially, it seemed my one-month-old headset to me. It refers to the design of a route for a scientific space which direction) they will expand and set range limits for was already broken. Further investigation revealed that mission, using the varying gravitational fields of the planets this expansion. For example, the flutist might set the range the most popular track —4’33’’ —was a three-movement to find pathways that are both scientifically interesting and of pulse values so that it expands to incorporate whole composition for any instrument where the score explicitly energy efficient. Orbit design seemed like a good model notes (as well as the initial quarter and half notes). The instructs performers not to play for the duration of four because it deals with the physical relationships of celestial performer might allow the pitch range to expand to all minutes and 33 , the listener’s experience consist - bodies, where Newton’s gravitational law is expressed chromatic notes between c3 and c4 inclusive, and could ing merely of ambient sounds in the surrounding environ - through their chaotic dynamical interaction. also decide that the volume will remain constant, at mp . In ment. Composed in 1952, this ostensibly silent work Of course, besides the mathematical imperatives men - this case, volume would be considered an inert parameter. caused an uproar. Around this time in his career, Cage tioned above, John Cage was looming. I decided to Finally, the performers decide in which order the param - began to incorporate chance elements—chance opera - embrace his ghostly presence not only in spirit but also in eters will expand. Only one parameter at a time can be tions and chance techniques—into his compositions. form. Cage wrote many compositions for open instrumen - dynamic, so that the players can concentrate on tracking its Here was the perfect inspiration for symplectic music. tation (including 4’33” ). I was drawn to this concept for expansion. The performer then remains at the outer limit Chance, chaos, stability, and comfort zone: these were the “The Symplectic Piece,” since it would inject an element of each completed parameter while expanding the follow - connections. Symplectic dynamics studies dynamical sys - of controlled unpredictability—or “deterministic chaos”— ing one, and so on, until the end of the piece. After all the tems, which in general show a whole spectrum of behav - into the performance. While an undergrad at Yale, I played parameters, their ranges, and the order of expansion have iors from stable motions to deterministic chaos—from the several of Cage’s Variations compositions (1958–67), in been determined, the performers assemble to try them out “comfort” of stability, the system evolves to be governed which the performers are given instructions for producing together. After a first playing/hearing, each performer completely by chance. sound, but no actual notation of rhythm, pitch, or any - makes adjustments to the scope of their own material, in Certain simple-looking systems exhibit interesting thing else. Though their actual manifestations could be order to complement their collaborators’ particular design. complex dynamics. The aforementioned movement of a compared to other 1960s “happenings,” the formal struc - The most improvisatory, or “chaotic,” element of the satellite in the gravitational field of the earth and the ture of the Variations tends to be more rigorous and specif - performance is orbiting . Orbiting requires a relationship with moon is one of them. The satellite can exhibit stability if ic. Something similar seemed suited to my needs. If I the other players, which involves hearing and interacting parked with the right momentum at one of the triangular tweaked the approach, I could produce a kind of algo - with what the others play from a vantage point that changes Lagrange points. On the other hand, the structure of the rithm—or “game”—that would allow musicians to choose (Continued on page 13)

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Finding Structure in Big Data

BY ANKUR MOITRA responding item. What is the structure in this data? This is some property of the problems that we actually want to ultimately what we hope to use to make good recommen - solve that makes them easier. In another work, we found a ow do we navigate the vast amount of data at our dis - dations. The basic idea is that a common interest is defined condition, which has been suggested within the machine Hposal? How do we choose a movie to watch, out of by a set of users (who share this interest) and a set of items. learning community, that makes these problems much eas - the 75,000 movies available on Netflix? Or a new book to And we expect each customer to have bought many items ier than in the worst case. 4 The crux of the assumption is read, among the 800,000 listed on Amazon? Or which in the set. We will call this a combinatorial rectangle (see that for every “interest,” there must be some item that (if news articles to read, out of the thousands written every - image). The basic hypothesis is that the entire table of data you buy it) is a strong indicator of your interest. For exam - day? Increasingly, these tasks are being delegated to com - we observe can be “explained” as a small number of these ple, whoever buys a top-of-the-line skillet is probably puters—recommendation systems analyze a large amount of rectangles. So in this table containing information about interested in cooking. This assumption is known in the data on user behavior, and use what they learn to make millions of items and millions of users, we hope to “explain” machine-learning as separability .5 In many personalized recommendations for each one of us. the behavior of the users by a small number of rectangles— instances of real data, practitioners have observed that this In fact, you probably encounter recommendation sys - each representing a common interest. condition is met by the parameters that their algorithm tems on an everyday basis: from Netflix to Amazon to finds. And what we showed is that under this condition, News, better recommendation systems translate to there are simple, fast algorithms that provably compute a a better user experience. There are some basic questions nonnegative factorization. we should ask: How good are these recommendations? In In fact, this is just one instance of a broader agenda: I fact, a more basic question: What does “good” mean? And believe that exploring these types of questions will be an how do they do it? As we will see, there are a number of important step in building bridges between theory and interesting mathematical questions at the heart of these practice. Our goal should not be to find a theoretical issues—most importantly, there are many widely used framework in which recommendations (and learning, A R T I

algorithms (in practice) whose behavior we cannot O more generally) are computationally hard problems, but M R

explain. Why do these algorithms work so well? Obvious - U rather one in which learning is easy—one that explains K N ly, we would like to put these algorithms on a rigorous the - A (for example) why simple recommendation systems are so oretical foundation and understand the computational If two customers have a common interest in cooking, Amazon can good. These questions lie somewhere between statistics complexity of the prob lems they are trying to solve. use information about which items one of them has bought to make and computer science, because the question is not : How Here, I will focus on one running example and use this good recommendations to the other and vice-versa. Ankur Moitra much data do you need to make good recommendations to explain the basic problems in detail, and some of the is trying to develop rigorous theoretical foundations for widely used (e.g., the statistical efficiency of an estimator)? Algorithms mathematical abstractions. Consider the case of Amazon. algorithms whose behavior we cannot explain. that use the bare minimum amount of data are all too I have purchased some items on Amazon recently: a fancy often very hard to compute. The emerging question is: set of cutting knives and a top-of-the-line skillet. What The fundamental mathematical problem is: If the data What are the best tradeoffs between making the most of other products might I be interested in? The basic tenet of can be “explained” by a small number of rectangles, can we your data, and running in some reasonable amount of designing a recommendation system is that the more data find them? This problem is called nonnegative matrix factor - time? The mathematical challenges abound in bringing you have available, the better your recommendations will ization , and it plays a large role in the design of real rec - these perspectives into not just recommendation sys - be. For example, Amazon could search through its vast ommendation systems. 1 In fact, there are man y algorithms tems—but into machine learning in general. I collection of user data for another customer (Alex) who that work quite well in practice (on real data). But is there has purchased the same two items. We both bought knives an efficient algorithm that provably works on every input? Ankur Moitra is an NSF Computing and Innovation and a skillet, and Amazon can deduce that we have a com - Recently, we showed that the answer is yes! 2 Fellow in the School of Mathematics at the Institute. mon interest in cooking. The key is: perhaps Alex has Our algorithm is based on a connection to a purely His primary research interests are in algorithms, bought another item, say a collection of cooking spices, algebraic question: Starting with the foundational work of learning, and convex geometry. Prior to joining IAS, and this is a good item to recommend to me, because I am Alfred Tarski and Abraham Seidenberg, a long line of he received his Ph.D. in 2011 and his M.S. in 2009 also interested in cooking. So the message is: lots of data research has focused on the task of deciding if a system of from the Massachusetts Institute of Technology, both (about similar customers) helps! polynomial inequalities has a solution. This problem can in theoretical computer science. Of course, Amazon’s job is not so easy. I also bought a be solved efficiently provided the number of distinct vari - Kindle. And what if someone else (Jeff) also bought a Kin - ables is small. 3 And indeed, whether or not our table of dle? I buy math books online, but maybe Jeff is more of a data has a “good” nonnegative matrix factorization can be 1 “Learning the Parts of an Object by Nonnegative Matrix Factorization,” Daniel Lee and H. Sebastian Seung, Nature 401, October 21, 1999 Harry Potter aficionado. Just because we both bought the rephrased equivalently as whether or not a certain system 2 “Computing a Nonnegative Matrix Factorization––Provably,” Sanjeev same item (a Kindle) does not mean that you should rec - of polynomial inequalities has a solution. So if our goal is Arora, Rong Ge, Ravi Kannan, and Ankur Moitra, Symposium on ommend Harry Potter books to me, and you certainly to design fast algorithms, the operative question is: Can we Theory of Computing , 2012 would not want to recommend math books to Jeff! The reduce the number of variables? This is precisely the route 3 “On the Computational Complexity and Geometry of the First-Order key is: What do the items I have purchased tell Amazon we took, and it led us to a (much faster) provable algo - Theory of the Reals,” James Renegar, Journal of Symbolic Computation about my interests ? Ideally, similar customers help us iden - rithm for nonnegative matrix factorization whose running 13: 3, March 1992 tify similar products, and vice-versa. time is optimal under standard complexity assumptions. 4 “Learning Topic Models––Going Beyond SVD,” Sanjeev Arora, Rong Ge, and Ankur Moitra, http://arxiv.org/abs/1204.1956, 2012 So how do they do it? Typically, the first step is to form Another fundamental mathematical question is: Can 5 “When does Nonnegative Matrix Factorization give a Correct Decom - a big table—rows represent items and columns represent we give a theoretical explanation for why heuristics for position into Parts?” David Donoho and Victoria Stodden, Advances in users. And an entry indicates if a customer bought the cor - these problems work so well in practice? There must be Neural Information Processing Systems 16, 2003

SYMPLECTIC PIECE (Continued from page 12) with time. As the performers develop and become comfort - (ABC) >> (AB)—C >> (ABC) >> A—B—C >> blissful. The players performed with formidable concen - able with their own chosen material, it becomes easier to (AC)—B >> (AB)—C >>, etc. tration, and afterward they expressed deep satisfaction move in and out of each other’s orbits. Such a “sentence” could map one realization of a per - with the opportunity to build their own compositions Over the course of our discussions, Helmut pointed out formance of Orbit Design . Of course, there are many, many from “scratch.” that the notion of orbiting around something is not exact - others. Indeed, as a consequence of the “open” nature of the As the date of the premiere grew near, I decided that ly defined in physics. Nonetheless, if the bodies’ masses score (which, as noted, consists only of written instruc - there was no reason to restrict Orbit Design to musical were about the same size, they could be said to orbit each tions), each performance necessarily ends up being distinct. realizations, as the instructions could apply equally well to other. He also drew a chart, which showed that if three Recognizing the unique element, the festival directors dancers, actors, jugglers, or any kind of performer. My celestial bodies A, B, C exist, they could be located either scheduled the premiere on three successive nights with friend, choreographer Abigail Levine, who had helped me close to each other (ABC), far away from each other (A— three completely different ensembles. On the first night, hone some of the formal concepts, attended the perform - B—C), or in various other configurations: (AB)—C, Orbit Design was performed by a percussion sextet featur - ances and decided to choreograph the work for an upcom - (BC)—A, (AC)—B. Also, if B and C were of comparable ing So Percussion Quartet, Doug Perkins, and Bobby Pre - ing project with Movement Research at the Judson size and A was much bigger, then the system (BC) could be vite; the second night it was played by the trio Forbidden Church in New York City. She paired it with a dance for said to orbit around A, and so forth. I felt that these max - Flute; and the third night it was realized by the string three performers, titled Distance Measures , and I partici - ims could apply directly to performers in Orbit Design . quartet Brooklyn Rider. The three incarnations differed pated as one of the musicians, alongside Forbidden Flute. Helmut had also noted that solutions exist to the vastly; each had its own profile and dynamic shape. The Thus the gravitational “choreography” to which Helmut three-body problem in which the bodies move from one percussion was sprawling yet serene, the flutes sporadic alluded finally had a chance to manifest itself in real time configuration to another in random order. For example: and sinuous, the string quartet alternately comic and and evolve into another . I

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Andrew W. Mellon Foundation Provides Critical Support for Assistant Professors

he Institute has received a $3 million grant from the Andrew W. Mellon Foundation diverse range of fields of historical study. “My time at the IAS was one of the most intel - Tto support one-year fellowships for assistant professors in the Institute’s School of His - lectually stimulating and productive periods of my life,” said Lauren Minsky, Assistant torical Studies. The donation will be matched by $3 million funded from the $100 mil - Professor at New York University in Abu Dhabi and former Member (2009–10) in the lion challenge grant provided by the Simons Foundation and the Charles and Lisa School of Historical Studies. “My fellowship allowed me the crucial opportunity to Simonyi Fund for Arts and Sciences to initiate the Institute’s current $200 million cam - research, think, read, and reflect for a full year without the teaching and administrative paign to strengthen its endowment. This will create a $6 million Andrew W. Mellon Fel - commitments that come with regular academic life. I also had the invaluable opportu - lowships for Assistant Professors Fund to provide a permanent endowment for fellowships nity to interact with other scholars from a wide range of fields and disciplines as col - to enable scholars to work at the Institute at a critical point in their careers. leagues and friends. It is impossible to fully express just how important and meaningful The Mellon Foundation has been funding fellowships at the Institute for historians in this opportunity was.” the early stages of their careers since 1996. The new grant will provide stable and secure The Andrew W. Mellon Foundation (www.mellon.org), formed in 1969 by the con - funding for the program, enabling future generations of scholars to benefit from the Insti - solidation of foundations established by the children of Andrew W. Mellon, supports five tute’s distinct environment, where they are free to pursue long-term goals away from the core program areas: higher education and scholarship; scholarly communications and teaching and administrative demands of university positions, and to interact with the information technology; art history, conservation, and museums; the performing arts; Institute’s permanent Faculty and other more senior colleagues. and conservation and the environment. The Foundation develops thoughtful, long-term Since the inception of the program, some forty-four scholars have benefited from the collaborations with grant recipients and invests sufficient funds for an extended period Mellon Foundation fellowships. These fellows have come from many institutions and a to accomplish the purpose at hand and achieve meaningful results. I

ANDREW W. MELLON FOUNDATION FELLOWSHIPS FOR ASSISTANT PROFESSORS, 2011–12

Yair Mintzker, Assistant Professor of European History, known literature from that period, which had been reconstructed in the Imperial Princeton University Library in the late first century B.C.E. Centering on a detailed and comprehensive study of the material and textual properties of a 300 B.C.E. bamboo manuscript from Supported by a Mellon fellowship, I spent this year working on the Southern Chinese state Chu, as well as some other manuscripts of similar prove - two separate projects. The first was a book about German city nance, the book demonstrates that early Chinese literature did not only consist of lin - walls. The early modern German city was legally, politically, and ear, logically consistent texts. One manuscript could combine different types of texts, culturally defined through its borders—the city walls. While e.g., a coherent argument could be supplemented with didactic catalogues that merely around 1700 practically all German cities were surrounded by a functioned as memory pegs, referring the reader in an indexical manner to previous wall, little over a century later very few still had one. The politi - instructions outside the text. In other cases, written texts merely functioned as reposi - cal causes and implications of the dramatic and often traumatic tories of textual material, often variants of the same content. Only in later redactions demolition of these centuries-old borders are the topic of my book “The Defortification were such texts reinterpreted and shaped into the coherent linear texts, which were of the German City, 1689–1866” (forthcoming from Cambridge University Press), which then transmitted to us through the two millennia of the Chinese empire. I completed while at the Institute. The second project is in its early stages. Tentatively called “The Multiple Deaths Vimalin Rujivacharakul, Associate Professor, Architecture of ‘Jew Süss,’” it is a new history of one of the most famous (and horrific) trials in eigh - and Critical Architectural Historiography, University of teenth-century Germany: the trial of Joseph Süss Oppenheimer, a Court Jew in Delaware Stuttgart, who was arrested, put on trial, and finally executed in February 1738. Since the sources about Oppenheimer’s trial are often contradictory, the book offers four dif - Holding an Andrew W. Mellon Fellowship for Assistant Pro - ferent versions of what happened during the trial—a Rashomon -like history that fessors at the Institute for Advanced Study is both a privilege encourages the reader to choose between four different historical accounts of the same and an opportunity to do things that are otherwise difficult to event. My year at the Institute provided me with the time and resources to work on achieve. My project focuses on an unusual subject—a set of these projects. I especially appreciated the opportunity to exchange ideas with IAS pictorial diaries of a Japanese architectural historian who Faculty and fellow Members. The experience was rewarding, humbling, and incredi - traveled the world at the turn of the twentieth century. The bly inspiring all at the same time. project’s scope is ambitious and its layers of cultural and historical elements compli - cate its context. Moreover, because I approach my subject through an unorthodox Matthias Richter, Assistant Professor of Chinese, University method that juxtaposes visual representation against written texts, I place my project of Colorado at Boulder squarely at the crossroads of intellectual history and architectural history. A transdis - Based on several years of previous research, I have been able to ciplinary project like this is normally a challenge to launch, especially for an early- use the two terms of my stay at the Institute to complete a book career scholar with a full-time teaching responsibility. However, not only did the manuscript and submit it to the publisher. The book “Reincar - Mellon fellowship enable me to take a yearlong leave to work on this project, it also nating the Disembodied Text: Textual Identity in Early Chinese allowed me to gain access to excellent research resources. I have been inspired by ideas Manuscripts” addresses the important methodological issue of garnered at seminars and lunchtime conversations; some sparked new perspectives how we can interpret the newly discovered manuscripts from and some redefined my original thinking. This is a place where I have friends, research pre-imperial China, i.e., before the late third century B.C.E., support, and superb intellectual stimulation—an ideal research community. without assimilating them anachronistically to the hitherto

PLANNED GIVING Daniel H. Saracino Endowment Supports Visiting Professorship in School of Mathematics

bequest made by Daniel H. Saracino, the Neil R. Grabois Professor of Mathematics Saracino’s current research interests include model theory—especially model theoretic A at Colgate University and former Visitor (1986) in the Institute’s School of Mathe - algebra (existentially complete, quantifier-eliminable, and homogeneous structures in matics, will enable the endowment of a Visiting Professorship in the School. The Daniel algebra) —and combinatorics. He earned his Ph.D. from Princeton University in 1972, H. Saracino Endowment will provide the School with the ability to invite a senior math - and taught at Yale and Wesleyan before coming to Colgate. The recipient of Colgate’s ematician for a year or more to work with the School’s Faculty and Members in an environ - Alumni Corporation Award for Distinguished Teaching (2003) and Phi Eta Sigma ment devoted to fundamental research across many areas of , theoretical Professor of the Year (1995), Saracino has been recognized throughout his career for his computer science, mathematical physics, and applied mathematics. Saracino was prompted outstanding teaching and mentoring, and was recently included in Princeton Review’s to include the Institute in his estate plans out of his deep admiration and respect for the The Best 300 Professors . I mission and work of the School and its Faculty, and in fond acknowledgment of his time as a Visitor. A planned gift may be tailored to meet a donor’s wishes and interests and help the “The Institute epitomizes excellence in mathematics, and I appreciate the way in Institute achieve its goals. It is hoped that former Members and Visitors will consider which the School of Mathematics makes it possible for its Members and Visitors to the Institute in the same way in which they would their alumni institutions, and devote themselves to their research in an ideal atmosphere, free of distractions and other plan accordingly. To learn more about what a planned gift can accomplish for the responsibilities,” said Saracino. “My bequest to the School provides a way of doing what donor and the Institute for Advanced Study, please contact Catie Newcombe, I can to further its work.” Senior Development Officer, at [email protected] or (609) 951-4542.

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Benedict H. Gross Elected to the Board of Trustees he Institute has ap - Tpointed Benedict H. Gross, the George Vasmer Leverett Profes - Friends Bolster Campaign to Raise Capital Funds sor of Mathematics at he $200 million Campaign for the Institute, initiated in 2011 with a $100 million challenge grant from the Simons Foun - Harvard University, to Tdation and the Charles and Lisa Simonyi Fund for Arts and Sciences, is well underway and will continue through June its Board of Trustees, ef - 30, 2015. With major pledges from the Leon Levy Foundation (see page 1), the Starr Foundation (see page 3), the Andrew fective July 1, 2012. W. Mellon Foundation (see page 14), and others, the total raised now stands at more than $48 million, almost halfway to - Gross was nominated by ward qualifying for the $100 million challenge grant. Support from the entire Institute community will be essential to com - the Institute’s School of plete the Campaign, which is aimed at strengthening the endowment on which the Institute relies heavily—covering as Mathematics and suc - much as 80 percent of annual operating expense in recent years. Benedict H. Gross ceeds Andrew J. Wiles, The Friends of the Institute, whose generosity has made an indelible impact on the Institute’s financial well-being over Royal Society Research the past three decades by providing IAS with its most significant source of unrestricted financial support, have joined the Professor at the University of Oxford, who had effort to raise capital funds for the Campaign. Building on years of success securing annual funds from Friends (over $778,000 served on the Institute’s Board since 2007. was raised in 2011 –12), Chair Jack Kerr, former Chairs Carolyn Sanderson and John Rassweiler, and members of the Devel - Gross, whose research is in number theory, opment Committee have been working to engage Friends in the Campaign, with great results. Friends have already com - received his undergraduate degree from Harvard mitted more than $500,000 in gifts and pledges and the Committee is looking to increase that sum significantly, as all Friends in 1971, a master’s degree from the University of are given the chance to participate in the Campaign over the next three years. Oxford in 1974, and his doctorate from Harvard “We want to see this unique opportunity for scholars continue into the future, building on the Institute’s phenomenal success in 1978. He served as Assistant Professor at since its founding,” notes Kerr and his wife Nora, Friends since 1998. “Access to the Institute enriches our lives in many ways. Princeton University from 1978–82, Associate There are frequent opportunities open to the Friends to participate in the life of the Institute, whether at formal lectures about Professor at Brown University from 1982–85, and current research, informal lunches with Members in the dining hall, or the numerous social events organized by the Friends. Professor at Harvard beginning in 1985. He was The scholars in this commu - Dean for Undergraduate Education at Harvard nity are very approachable from 2002–03 and Dean of Harvard College from and love to talk about their 2003–07; in these positions, he oversaw the first work, which is always interest - review of undergraduate education at Harvard in ing. It’s a terrific opportunity nearly thirty years. to engage the brightest minds The American Mathematical Society awarded of our generation.” Gross and his collaborators the Frank Nelson To learn more about the Cole Prize in Number Theory in 1987 for work on Campaign for the Institute, the L-functions of elliptic curves. Among other please contact Michael Gehret, honors, he is the recipient of a MacArthur Fel - Associate Director for Devel - lowship and a member of the American Academy opment and Public Affairs, at of Arts and Sciences and the National Academy (609) 734-8218 or mgehret@ of Sciences. ias.edu. To learn more about the Gross is the author, with Joe Harris, of The Friends, please contact Pamela Magic of Numbers (Prentice Hall, 2003), which Hughes, Senior Development introduces non-mathematicians to the mathe - Officer and Friends Liaison, at Executive Committee of the Friends in May, front row: Cynthia Hillas, Vicky Corrodi, Emily matical mode of thought. A course on abstract (609) 734-8204 or phughes@ Firmenich, Peter Goddard (then Director of the Institute), Carolyn Sanderson, Debbie Lunder, algebra taught by Gross is available at www.exten- ias.edu, or visit the Friends Francesca Liechenstein, Tina Greenberg, Florence Kahn, Vivian Shapiro; back row: Jack McCarthy, sion.harvard.edu/open-learning-initiative/ab- pages on the Institute’s website, Lew Maltby, Jack Kerr, John Haines, John Rassweiler, Weezie Steffens, Michael Morandi, Luke stract-algebra. I www.ias.edu/people/friends. I Visconti, Lynn Johnston, Martin Chooljian, Brig Gebert

LIBERAL DEMOCRATIC LEGACIES (Continued from page 11)

cultural refinement could take part in this aristocracy. In for Haykal and his liberal democratic orientation. that the national struggle would be lengthy and that contrast to Husayn and his activity within the popular- ______British colonial rule would last until the late 50s. It oriented Wafd, Haykal was the intellectual leader of the became increasingly difficult for them to overcome a Liberal Constitutionalist Party, which promoted a he defense of liberal democracy by Haykal, Husayn, glaring obstacle: the fact that the very European West - patronizing and paternalistic liberalism and was largely a Tand many of these intellectuals suffered from limita - ern culture that provided concepts and models for liber - continuation of the early Hizb al-Umma and al-Jarida . tions, two of which I’d like to emphasize. The first is that al democratic ideas and institutions was simultaneously While Husayn viewed himself as a challenger of this the parliamentary system did not function smoothly the imperial power that threatened their local culture. school, Haykal viewed himself as a guardian of its social throughout the entire period. In particular, the 1930s, As newer and more radical forces emerged in the cultur - philosophy. Haykal assumed that the masses would fol - ’40s, and early ’50s saw episodes of bitter conflict between al and political arena, they rejected liberalism as a for - low the charismatic power of the prophetic philosopher- contending forces that resulted in occasional constitu - eign imperialist institution. Worse yet, they claimed that kings and would accept their authority as the tional adjustments, which impeded parliamentary per - Western liberal democracy’s purported freedom was a embodiment of the collective general will. Thus, the formance. While the monarch conspired to undermine tool in the service of a narrow Western elite promoting masses would be led to a modern democratic system the democratic government and to guide it toward a more Western colonial discourse, which usurped the freedom “unaware” and without really understanding its essence. authoritarian orientation, radical extra-parliamentary of the masses through its sham parliamentary system. However, at critical junctures, Haykal also demonstrat - forces, led by Young Egypt and to a lesser extent also by A newer radicalized aimed at decolonial - ed that he was in fact a liberal democrat. In the early 1930s the Society of the Muslim Brothers, strove to delegitimize ization of Egypt and the Arab Middle East undermined when the dictatorial regime of Isma‘il Sidqi (1930–33) its very existence. British colonial intervention in poli - the intellectual project to institutionalize a liberal demo- undermined the 1923 constitution and thereby threatened tics similarly hindered its ability to function. cracy in Egypt and the Arab world. The Free Officers’ to destroy the parliamentary government, Haykal led a The second limitation was more substantial. It coup d’etat of July 1952 and the emergence of the author - democratic struggle against Sidqi. In his book al-Siyasa al- involved the question: to what extent was such an aim itarian republican regime abruptly ended this project. Misriyya wa al-Inqilab al-Dusturi (Egyptian Politics and the realizable within the colonial or semi-colonial context? However, from a historical perspective, the intellectuals’ Constitutionalist Coup ), published in 1931, Haykal waged a Can intellectuals promote a true liberal democracy in a liberal democratic legacy has proven to be enduring. Their fierce and unrelenting defense of democracy from any colonized environment? To be sure, the intellectuals liberal texts were canonized and remained popular years authoritarian options of autocracy or dictatorship. whom I’ve discussed also grappled with these questions, after their deaths. They were continuously consumed by The collapse of Sidqi’s authoritative government and and concluded that it was feasible. They assumed that successive generations. To what extent this legacy may the restoration of the constitution and the reassertion of their efforts were an integral element in Egypt’s struggle serve as a source of inspiration and legitimization in cur - parliamentary life toward the mid-1930s were a triumph for liberation from colonial rule. They did not anticipate rent conditions in Egypt remains to be seen. I

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ng nar, Higgs presented his theory of how most fundamental particles acquire mass as they interact with an energy J dv 0

e field that exists everywhere in the universe. Soon after the CERN announcement, Dyson sent a message to Higgs a 8 o n 5

f through a colleague, David Derbes: “Please give my warm congratulations to Peter if you see him. What he did 4 ce a 0 d d was beautiful and will outlast the temporary flood of hype.” Derbes met Higgs for dinner in Glasgow on July 24, St r and reported back to Dyson: “I think he was very appreciative.” e udy s

s. After nearly five decades of searching for of the existence of the Higgs particle that would result if Higgs’s theory is correct, Faculty and Members—along with Robbert Dijkgraaf on his fourth day as the ninth Director of the Institute—attended a pre-dawn celebration in Bloomberg Hall to witness the announcement by Rolf Heuer, Director-General of CERN (page 4). The news was momentous. Faculty and Members at the Insti - tute have contributed many of the theoretical foundations of the Standard Model, which describes the funda - mental building blocks of the universe, and its possible modifications. The discovery of a particle that looks much like the Higgs —the final element of the Standard Model that needs to be confirmed experimentally —promises to further understanding of the origin of mass and help clarify some long-standing mysteries. The late Marston Morse, Professor in the School of Mathematics, believed that mathematicians, physicists, and artists are bound through a willingness to admit uncertainty and to try to understand it. In the following pages, John Hopfield , Visiting Professor in the School of Natural Sciences, writes about wanting to understand how mind emerges from brain (page 1); Helmut Hofer , Professor in the School of Mathematics, and Derek Bermel , Artist-in- Residence, find a common ground between symplectic dynamics, music, and John Cage (page 1); Brandon Look ,

Princ eton, Permi t U. S. Non-Prfi t Member in the School of Historical Studies, delves into the likelihood of understanding, from a metaphysical per - spective, why there is something rather than nothing (page 5); and Richard Taylor , Professor in the School of Math -

Post age ematics, explains the curiosity and beauty of modular arithmetic, which has led to real-world applications, such as

#4 9 the way DVDs store or satellites transmit large amounts of data without corrupting it (page 6). Morse proposed a psy - chological and spiritual affinity between these types of explorations pursued by the individual who “chooses one NJ Org. pattern for beauty’s sake, and pulls it down to earth, no one knows how. Afterwards the logic of words and of forms PAI D sets the pattern right. Only then can one tell someone else. The first pattern remains in the shadows of the mind.”